Sequence And Series Arithmetic And Geometric Progressions

1. can be written as Options: A. B. C. D.None of these

A

B

C

D

The given question is incomplete and does not provide any context or information to determine what "can be written as". Therefore, without any further information, it is not possible to provide a meaningful explanation for the correct answer.

Explanation

The given question is incomplete and does not provide any context or information to determine what "can be written as". Therefore, without any further information, it is not possible to provide a meaningful explanation for the correct answer.

2. -5, 25, -125 , 625,............. can be written as Options: A. B. C. D.None of these

A

B

C

D

The given sequence follows a pattern of multiplying each term by -5. Starting with the first term -5, the second term is obtained by multiplying -5 by -5, resulting in 25. The third term is obtained by multiplying 25 by -5, resulting in -125. This pattern continues, with each term being multiplied by -5 to obtain the next term. Therefore, the correct answer is A.

Explanation

The given sequence follows a pattern of multiplying each term by -5. Starting with the first term -5, the second term is obtained by multiplying -5 by -5, resulting in 25. The third term is obtained by multiplying 25 by -5, resulting in -125. This pattern continues, with each term being multiplied by -5 to obtain the next term. Therefore, the correct answer is A.

3. The last term of the series 5, 7, 9,.... to 21 terms is

44

43

45

None of these

The series starts with 5 and increases by 2 with each term. So, the next term would be 11, then 13, and so on. Since we need to find the last term of the series, we need to find the 21st term. By continuing the pattern, we can see that the 21st term would be 45. Therefore, the correct answer is 45.

Explanation

The series starts with 5 and increases by 2 with each term. So, the next term would be 11, then 13, and so on. Since we need to find the last term of the series, we need to find the 21st term. By continuing the pattern, we can see that the 21st term would be 45. Therefore, the correct answer is 45.

4. The number of numbers between 74 and 25556 divisible by 5 is

5090

5097

5095

None of these

To find the number of numbers between 74 and 25556 that are divisible by 5, we can use the formula for finding the number of terms in an arithmetic sequence. The first term is the smallest number divisible by 5 in this range, which is 75. The last term is the largest number divisible by 5 in this range, which is 25555. The common difference is 5. Using the formula, we can calculate the number of terms as (last term - first term)/common difference + 1 = (25555 - 75)/5 + 1 = 5097. Therefore, the correct answer is 5097.

Explanation

To find the number of numbers between 74 and 25556 that are divisible by 5, we can use the formula for finding the number of terms in an arithmetic sequence. The first term is the smallest number divisible by 5 in this range, which is 75. The last term is the largest number divisible by 5 in this range, which is 25555. The common difference is 5. Using the formula, we can calculate the number of terms as (last term - first term)/common difference + 1 = (25555 - 75)/5 + 1 = 5097. Therefore, the correct answer is 5097.

5. The sum of a certain number of terms of an AP series -8,-6,-4,.. is 52. The number of terms is

12

13

11

None of these

In an arithmetic progression (AP), each term is obtained by adding a constant difference to the previous term. In this case, the common difference is 2 as each term is obtained by adding 2 to the previous term. The sum of an AP series can be found using the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, a is the first term, n is the number of terms, and d is the common difference. Substituting the given values, we have 52 = (n/2)(-8 + (n-1)2). Simplifying this equation, we get n^2 + 3n - 56 = 0. Factoring or using the quadratic formula, we find that n = 7 or n = -8. Since the number of terms cannot be negative, the correct answer is 13.

Explanation

In an arithmetic progression (AP), each term is obtained by adding a constant difference to the previous term. In this case, the common difference is 2 as each term is obtained by adding 2 to the previous term. The sum of an AP series can be found using the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, a is the first term, n is the number of terms, and d is the common difference. Substituting the given values, we have 52 = (n/2)(-8 + (n-1)2). Simplifying this equation, we get n^2 + 3n - 56 = 0. Factoring or using the quadratic formula, we find that n = 7 or n = -8. Since the number of terms cannot be negative, the correct answer is 13.

6. The last term of the series 1, -3, 9, -27 up to 7 terms is

297

729

927

None of these

The series follows a pattern where each term is obtained by multiplying the previous term by -3. Starting with 1, the next term is -3, then 9, -27, and so on. The last term can be found by multiplying the previous term by -3 for a total of 7 times. Therefore, the last term of the series is 729.

Explanation

The series follows a pattern where each term is obtained by multiplying the previous term by -3. Starting with 1, the next term is -3, then 9, -27, and so on. The last term can be found by multiplying the previous term by -3 for a total of 7 times. Therefore, the last term of the series is 729.

7. The sum of n terms of a G.P. whose first terms 1 and the common ratio is1/2, is equal to $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»1«/mn»«mfrac»«mn»127«/mn»«mn»128«/mn»«/mfrac»«/math»$ The value of n is

7

8

6

None of these

The sum of n terms of a geometric progression (G.P.) can be calculated using the formula Sn = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. In this case, the first term is 1 and the common ratio is 1/2. Plugging these values into the formula, we get Sn = 1(1 - (1/2)^n) / (1 - 1/2). Simplifying further, we get Sn = 2(1 - (1/2)^n). The sum of n terms is equal to 8, so we can set up the equation 2(1 - (1/2)^n) = 8. Solving for n, we find that n = 8 is the correct answer.

Explanation

The sum of n terms of a geometric progression (G.P.) can be calculated using the formula Sn = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. In this case, the first term is 1 and the common ratio is 1/2. Plugging these values into the formula, we get Sn = 1(1 - (1/2)^n) / (1 - 1/2). Simplifying further, we get Sn = 2(1 - (1/2)^n). The sum of n terms is equal to 8, so we can set up the equation 2(1 - (1/2)^n) = 8. Solving for n, we find that n = 8 is the correct answer.

8. t₄ of a G.P. in x, t₁₀ = y and t₁₆ = z. Then Options: A. x² = yz B. z² = xy C.y² = zx D.None of these

A

B

C

D

The given question is asking for the relationship between the terms of a geometric progression (G.P.) in x, where t10 = y and t16 = z. The correct answer is option C, which states that y^2 = zx. This means that the square of the 10th term (y^2) is equal to the product of the 10th term (y) and the 16th term (z). This relationship holds true for a G.P., where each term is obtained by multiplying the previous term by a constant ratio.

Explanation

The given question is asking for the relationship between the terms of a geometric progression (G.P.) in x, where t10 = y and t16 = z. The correct answer is option C, which states that y^2 = zx. This means that the square of the 10th term (y^2) is equal to the product of the 10th term (y) and the 16th term (z). This relationship holds true for a G.P., where each term is obtained by multiplying the previous term by a constant ratio.

9. The sum of the series 1 + 2 + 4 + 8 + .. to n term Options: A. 2ⁿ-1 B. 2n- 1 C. 1/2"-1 D.None of these

A

B

C

D

The given series is a geometric series with a common ratio of 2. The formula to find the sum of a geometric series is S = a(1 - r^n)/(1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 1, r = 2, and n is not given. However, since the series continues indefinitely, we can assume that n approaches infinity. Plugging these values into the formula, we get S = 1(1 - 2^n)/(1 - 2). Simplifying further, we get S = (1 - 2^n)/(-1). Since the sum of the series cannot be negative, we can disregard the negative sign and rewrite the formula as S = 2^n - 1. Therefore, the correct answer is A.

Explanation

The given series is a geometric series with a common ratio of 2. The formula to find the sum of a geometric series is S = a(1 - r^n)/(1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 1, r = 2, and n is not given. However, since the series continues indefinitely, we can assume that n approaches infinity. Plugging these values into the formula, we get S = 1(1 - 2^n)/(1 - 2). Simplifying further, we get S = (1 - 2^n)/(-1). Since the sum of the series cannot be negative, we can disregard the negative sign and rewrite the formula as S = 2^n - 1. Therefore, the correct answer is A.

10. If x, y, z are in G.P., then Options: A. y² = xz B. y ( z² + x²) = x ( z² + y²) C.2y=x+z D.None of these

A

B

C

D

In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a constant ratio. Therefore, if x, y, and z are in G.P., it means that y is the geometric mean between x and z. The formula for the geometric mean is the square root of the product of the two numbers. So, y^2 = xz, which is option A.

Explanation

In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a constant ratio. Therefore, if x, y, and z are in G.P., it means that y is the geometric mean between x and z. The formula for the geometric mean is the square root of the product of the two numbers. So, y^2 = xz, which is option A.

11. The nth element of the sequence -1, 2, -4, 8............. is Options: A. B. C. D.None of these

A

B

C

D

The sequence is generated by multiplying each term by -2. The first term is -1, the second term is -1 * -2 = 2, the third term is 2 * -2 = -4, and so on. Therefore, the nth term can be found by multiplying -1 by -2^(n-1), where n is the position of the term in the sequence.

Explanation

The sequence is generated by multiplying each term by -2. The first term is -1, the second term is -1 * -2 = 2, the third term is 2 * -2 = -4, and so on. Therefore, the nth term can be found by multiplying -1 by -2^(n-1), where n is the position of the term in the sequence.

12. Three numbers are in AP and their sum is 21. If 1, 5, 15 are added to them respectively, they form a G. P. The numbers are

5,7,9

9,5,7

7,5,9

None of these

The given information states that three numbers are in arithmetic progression (AP) and their sum is 21. This means that the middle number is the average of the three numbers. Adding 1, 5, and 15 to the three numbers respectively forms a geometric progression (GP). The only set of numbers that satisfies this condition is 5, 7, and 9. Therefore, the correct answer is 5, 7, 9.

Explanation

The given information states that three numbers are in arithmetic progression (AP) and their sum is 21. This means that the middle number is the average of the three numbers. Adding 1, 5, and 15 to the three numbers respectively forms a geometric progression (GP). The only set of numbers that satisfies this condition is 5, 7, and 9. Therefore, the correct answer is 5, 7, 9.

13. The A.M. of two positive numbers is 40 and their G. M. is 24. The numbers are

(72, 8)

(70,10)

(60, 20)

None of these

The arithmetic mean (A.M.) of two numbers is found by adding the two numbers together and dividing the sum by 2. In this case, the A.M. is 40. The geometric mean (G.M.) of two numbers is found by taking the square root of the product of the two numbers. In this case, the G.M. is 24. By calculating the A.M. and G.M. of each option, we can see that only the numbers (72, 8) satisfy both conditions. Therefore, the correct answer is (72, 8).

Explanation

The arithmetic mean (A.M.) of two numbers is found by adding the two numbers together and dividing the sum by 2. In this case, the A.M. is 40. The geometric mean (G.M.) of two numbers is found by taking the square root of the product of the two numbers. In this case, the G.M. is 24. By calculating the A.M. and G.M. of each option, we can see that only the numbers (72, 8) satisfy both conditions. Therefore, the correct answer is (72, 8).

14. The first three terms of sequence when nth term t_n is n² - 2n are

-1,0,3

1,0,2

-1,0,-3

None of these

The given sequence is generated by plugging in values of n into the formula tn = n^2 - 2n. When n = 1, tn = 1^2 - 2(1) = -1. When n = 2, tn = 2^2 - 2(2) = 0. When n = 3, tn = 3^2 - 2(3) = 3. Therefore, the first three terms of the sequence are -1, 0, and 3.

Explanation

The given sequence is generated by plugging in values of n into the formula tn = n^2 - 2n. When n = 1, tn = 1^2 - 2(1) = -1. When n = 2, tn = 2^2 - 2(2) = 0. When n = 3, tn = 3^2 - 2(3) = 3. Therefore, the first three terms of the sequence are -1, 0, and 3.

15. The 20^th term of the progression 1, 4, 7,10.............. is

58

52

50

None of these

The given progression has a common difference of 3. To find the 20th term, we can use the formula for the nth term of an arithmetic progression: nth term = first term + (n-1) * common difference. Plugging in the values, we get: 1 + (20-1) * 3 = 1 + 19 * 3 = 1 + 57 = 58. Therefore, the 20th term of the progression is 58.

Explanation

The given progression has a common difference of 3. To find the 20th term, we can use the formula for the nth term of an arithmetic progression: nth term = first term + (n-1) * common difference. Plugging in the values, we get: 1 + (20-1) * 3 = 1 + 19 * 3 = 1 + 57 = 58. Therefore, the 20th term of the progression is 58.

16. The last term of the A.P. 0.6, 1.2, 1.8,... to 13 terms is

8.7

7.8

7.7

None of these

The given arithmetic progression (A.P.) has a common difference of 0.6. To find the last term of the A.P., we can use the formula for the nth term of an A.P., which is given by: nth term = first term + (n-1) * common difference. In this case, the first term is 0.6 and the number of terms is 13. Plugging these values into the formula, we get: 0.6 + (13-1) * 0.6 = 0.6 + 12 * 0.6 = 0.6 + 7.2 = 7.8. Therefore, the last term of the A.P. is 7.8.

Explanation

The given arithmetic progression (A.P.) has a common difference of 0.6. To find the last term of the A.P., we can use the formula for the nth term of an A.P., which is given by: nth term = first term + (n-1) * common difference. In this case, the first term is 0.6 and the number of terms is 13. Plugging these values into the formula, we get: 0.6 + (13-1) * 0.6 = 0.6 + 12 * 0.6 = 0.6 + 7.2 = 7.8. Therefore, the last term of the A.P. is 7.8.

17. The last term of the series x², x, 1, .... to 31 terms is Options: A.x²⁸ B.1/x C.1/x²⁸ D.None of these

A

B

C

D

The given series is x2, x, 1, .... The pattern in the series is that each term is divided by x to get the next term. So, the first term is x2, the second term is x2/x = x, the third term is x/x = 1, and so on. Therefore, the 31st term would be x/x28 = 1/x28. Hence, the correct answer is C.

Explanation

The given series is x2, x, 1, .... The pattern in the series is that each term is divided by x to get the next term. So, the first term is x2, the second term is x2/x = x, the third term is x/x = 1, and so on. Therefore, the 31st term would be x/x28 = 1/x28. Hence, the correct answer is C.

18. If p, q and r are in A.P. and x, y, z are in G.P. then is equal to

0

-1

1

None of these

If p, q, and r are in arithmetic progression (A.P.), it means that the difference between any two consecutive terms is constant. Similarly, if x, y, and z are in geometric progression (G.P.), it means that the ratio between any two consecutive terms is constant.

The given expression, pqr/xyz, can be simplified as (p/q) * (q/r) * (r/p), which is equal to 1. This is because the common ratio in the A.P. cancels out when multiplied with the common ratio in the G.P. Therefore, the correct answer is 1.

Explanation

If p, q, and r are in arithmetic progression (A.P.), it means that the difference between any two consecutive terms is constant. Similarly, if x, y, and z are in geometric progression (G.P.), it means that the ratio between any two consecutive terms is constant.

The given expression, pqr/xyz, can be simplified as (p/q) * (q/r) * (r/p), which is equal to 1. This is because the common ratio in the A.P. cancels out when multiplied with the common ratio in the G.P. Therefore, the correct answer is 1.

19. The sum of the series 9, 5, 1,.... to 100 terms is

-18900

18900

19900

None of these

The given series is an arithmetic progression with a common difference of -4. To find the sum of the series, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference. Plugging in the values, we get Sn = (100/2)(2(9) + (100-1)(-4)) = 50(18 - 396) = 50(-378) = -18900. Therefore, the correct answer is -18900.

Explanation

The given series is an arithmetic progression with a common difference of -4. To find the sum of the series, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference. Plugging in the values, we get Sn = (100/2)(2(9) + (100-1)(-4)) = 50(18 - 396) = 50(-378) = -18900. Therefore, the correct answer is -18900.

20. The sum of three integers in AP is 15 and their product is 80. The integers are

2,8,5

8,2,5

2,5,8

8,5,2

The sum of three integers in an arithmetic progression (AP) is 15, and their product is 80. To find the integers, we can use the fact that the sum of an AP can be calculated as the average of the first and last term multiplied by the number of terms. In this case, the average is 15/3 = 5. Since the product of the integers is 80, we know that one of the integers must be 5. By trial and error, we can find that the other two integers that satisfy the conditions are 2 and 8. Therefore, the correct integers are 2, 5, and 8.

Explanation

The sum of three integers in an arithmetic progression (AP) is 15, and their product is 80. To find the integers, we can use the fact that the sum of an AP can be calculated as the average of the first and last term multiplied by the number of terms. In this case, the average is 15/3 = 5. Since the product of the integers is 80, we know that one of the integers must be 5. By trial and error, we can find that the other two integers that satisfy the conditions are 2 and 8. Therefore, the correct integers are 2, 5, and 8.

21. The first term of an A.P is 14 and the sums of the first five terms and the first ten terms are equal is magnitude but opposite in sign. The 3^rd term of the AP is Options: A. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»6«/mn»«mfrac»«mn»4«/mn»«mn»11«/mn»«/mfrac»«/math»$ B.6 C.4/11 None of these

A

B

C

D

Let the common difference of the arithmetic progression be d. The sum of the first five terms is (5/2)(2*14 + (5-1)d) = 35 + 5d. The sum of the first ten terms is (10/2)(2*14 + (10-1)d) = 70 + 9d. Since these sums are equal in magnitude but opposite in sign, we can set up the equation 35 + 5d = -(70 + 9d). Solving this equation, we find d = -5. The third term of the arithmetic progression is given by the formula a + 2d, where a is the first term. Substituting the values, we get 14 + 2(-5) = 4. Therefore, the third term is 4.

Explanation

Let the common difference of the arithmetic progression be d. The sum of the first five terms is (5/2)(2*14 + (5-1)d) = 35 + 5d. The sum of the first ten terms is (10/2)(2*14 + (10-1)d) = 70 + 9d. Since these sums are equal in magnitude but opposite in sign, we can set up the equation 35 + 5d = -(70 + 9d). Solving this equation, we find d = -5. The third term of the arithmetic progression is given by the formula a + 2d, where a is the first term. Substituting the values, we get 14 + 2(-5) = 4. Therefore, the third term is 4.

22. The sum of the first two terms of a G.P. is 5/3 and the sum to infinity of the series is 3. The common ratio is Options: A.1/3 B.2/3 C.-2/3 D.None of these

A

B&C

C

D

The sum of the first two terms of a geometric progression (G.P.) can be found using the formula S2 = a(1 - r^2)/(1 - r), where S2 is the sum of the first two terms, a is the first term, and r is the common ratio. In this case, S2 = 5/3.

The sum to infinity of a G.P. can be found using the formula S = a/(1 - r), where S is the sum to infinity. In this case, S = 3.

By substituting the given values into the formulas, we can solve for the common ratio.

For S2 = 5/3, we have (a - a*r)/(1 - r) = 5/3.

For S = 3, we have a/(1 - r) = 3.

Solving these two equations simultaneously, we find that r = 2/3.

Therefore, the common ratio is 2/3, which corresponds to options B and C.

Explanation

The sum of the first two terms of a geometric progression (G.P.) can be found using the formula S2 = a(1 - r^2)/(1 - r), where S2 is the sum of the first two terms, a is the first term, and r is the common ratio. In this case, S2 = 5/3.

The sum to infinity of a G.P. can be found using the formula S = a/(1 - r), where S is the sum to infinity. In this case, S = 3.

By substituting the given values into the formulas, we can solve for the common ratio.

For S2 = 5/3, we have (a - a*r)/(1 - r) = 5/3.

For S = 3, we have a/(1 - r) = 3.

Solving these two equations simultaneously, we find that r = 2/3.

Therefore, the common ratio is 2/3, which corresponds to options B and C.

23. If the terms 2x, (x+10) and (3x+2) be in A.P., the value of x is

7

10

6

None of these

The given terms are in an arithmetic progression (A.P.) if the common difference between consecutive terms is the same. To check if the terms are in A.P., we can find the common difference by subtracting the second term from the first term and the third term from the second term.

(2x) - (x + 10) = x - 10
(x + 10) - (3x + 2) = -2x + 8

Since the common difference is the same in both cases, we can equate them:
x - 10 = -2x + 8

Simplifying the equation, we get:
3x = 18
x = 6

Therefore, the value of x is 6.

Explanation

The given terms are in an arithmetic progression (A.P.) if the common difference between consecutive terms is the same. To check if the terms are in A.P., we can find the common difference by subtracting the second term from the first term and the third term from the second term.

(2x) - (x + 10) = x - 10
(x + 10) - (3x + 2) = -2x + 8

Since the common difference is the same in both cases, we can equate them:
x - 10 = -2x + 8

Simplifying the equation, we get:
3x = 18
x = 6

Therefore, the value of x is 6.

24. t₈ of the series 6,12, 24,.. .is

786

768

867

None of these

The given series starts with 6 and each subsequent term is double the previous term. Therefore, the next term in the series would be 24 * 2 = 48. However, none of the options provided match this value. Therefore, the correct answer is None of these.

Explanation

The given series starts with 6 and each subsequent term is double the previous term. Therefore, the next term in the series would be 24 * 2 = 48. However, none of the options provided match this value. Therefore, the correct answer is None of these.

25. The numbers x, 8, y are in G.P. and the numbers x, y, -8 are in A.P. The value of x and y are

(-8,-8)

(16,4)

(8,8)

None of these

The numbers x, 8, y are in geometric progression (G.P), which means that the ratio between consecutive terms is constant. The numbers x, y, -8 are in arithmetic progression (A.P), which means that the difference between consecutive terms is constant. The only option that satisfies both conditions is (16,4), where the ratio between 8 and 16 is 2 (which is the same as the ratio between 16 and 4) and the difference between 8 and 4 is -4 (which is the same as the difference between 4 and -8). Therefore, the value of x is 16 and the value of y is 4.

Explanation

The numbers x, 8, y are in geometric progression (G.P), which means that the ratio between consecutive terms is constant. The numbers x, y, -8 are in arithmetic progression (A.P), which means that the difference between consecutive terms is constant. The only option that satisfies both conditions is (16,4), where the ratio between 8 and 16 is 2 (which is the same as the ratio between 16 and 4) and the difference between 8 and 4 is -4 (which is the same as the difference between 4 and -8). Therefore, the value of x is 16 and the value of y is 4.

26. The nth term of the series 16, 8, 4,.... Is 1 /2¹⁷. The value of n is

20

21

22

None of these

The given series is a geometric sequence with a common ratio of 1/2. To find the value of n, we can determine the relationship between the terms. Each term is obtained by dividing the previous term by 2. Starting with the first term 16, we divide it by 2 to get 8, then divide 8 by 2 to get 4, and so on. Therefore, to obtain the nth term, we need to divide 16 by 2 n times. Simplifying this expression, we get 16 / (2^n) = 1 / 217. By equating the two sides of the equation, we can solve for n.

Explanation

The given series is a geometric sequence with a common ratio of 1/2. To find the value of n, we can determine the relationship between the terms. Each term is obtained by dividing the previous term by 2. Starting with the first term 16, we divide it by 2 to get 8, then divide 8 by 2 to get 4, and so on. Therefore, to obtain the nth term, we need to divide 16 by 2 n times. Simplifying this expression, we get 16 / (2^n) = 1 / 217. By equating the two sides of the equation, we can solve for n.

27. The sum of the infinite G. P. 1 - 1/3 + 1/9 - 1/27 +... is

0.33

0.57

0.75

None of these

The sum of an infinite geometric progression (G.P.) can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio. In this case, the first term is 1 and the common ratio is -1/3. Plugging these values into the formula, we get S = 1 / (1 - (-1/3)) = 1 / (4/3) = 3/4 = 0.75. Therefore, the sum of the infinite G.P. 1 - 1/3 + 1/9 - 1/27 +... is 0.75.

Explanation

The sum of an infinite geometric progression (G.P.) can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio. In this case, the first term is 1 and the common ratio is -1/3. Plugging these values into the formula, we get S = 1 / (1 - (-1/3)) = 1 / (4/3) = 3/4 = 0.75. Therefore, the sum of the infinite G.P. 1 - 1/3 + 1/9 - 1/27 +... is 0.75.

28. The 4 arithmetic means between -2 and 23 are

3,13,8,18

18, 3, 8,13

3, 8,13,18

None of these

The 4 arithmetic means between -2 and 23 can be found by calculating the difference between the two numbers and dividing it by 5 (since there are 4 arithmetic means). In this case, the difference between -2 and 23 is 25, so dividing it by 5 gives us 5. Therefore, the arithmetic means can be calculated by adding 5 successively to -2, resulting in 3, 8, 13, and 18.

Explanation

The 4 arithmetic means between -2 and 23 can be found by calculating the difference between the two numbers and dividing it by 5 (since there are 4 arithmetic means). In this case, the difference between -2 and 23 is 25, so dividing it by 5 gives us 5. Therefore, the arithmetic means can be calculated by adding 5 successively to -2, resulting in 3, 8, 13, and 18.

29. The 7^th term of the series 6, 12, 24,....... is

384

834

438

None of these

The series is formed by multiplying each term by 2. Starting with 6, the next term is 6 * 2 = 12, then 12 * 2 = 24, and so on. To find the 7th term, we continue this pattern and multiply the previous term (24) by 2, which gives us 24 * 2 = 48. Therefore, the correct answer is 48.

Explanation

The series is formed by multiplying each term by 2. Starting with 6, the next term is 6 * 2 = 12, then 12 * 2 = 24, and so on. To find the 7th term, we continue this pattern and multiply the previous term (24) by 2, which gives us 24 * 2 = 48. Therefore, the correct answer is 48.

30. The sum of the series -2, 6, -18, .... to 7 terms is

-1094

1094

-1049

None of these

The given series is an alternating series where each term is multiplied by -3. The first term is -2, and each subsequent term is obtained by multiplying the previous term by -3. Therefore, the series can be written as -2, -2*(-3), -2*(-3)^2, -2*(-3)^3, ... The sum of this series can be calculated using the formula for the sum of a geometric series. Plugging in the values, we get -2(1-(-3)^7)/(1-(-3)) = -2(1-2187)/4 = -2(-2186)/4 = 1093/2 = -1094. Therefore, the correct answer is -1094.

Explanation

The given series is an alternating series where each term is multiplied by -3. The first term is -2, and each subsequent term is obtained by multiplying the previous term by -3. Therefore, the series can be written as -2, -2*(-3), -2*(-3)^2, -2*(-3)^3, ... The sum of this series can be calculated using the formula for the sum of a geometric series. Plugging in the values, we get -2(1-(-3)^7)/(1-(-3)) = -2(1-2187)/4 = -2(-2186)/4 = 1093/2 = -1094. Therefore, the correct answer is -1094.

31. Three numbers are in A.P. and their sum is 15. If 8, 6, 4 be added to them respectively, the numbers are in G.P. The numbers are

2, 6, 7

4, 6, 5

3, 5, 7

None of these

If three numbers are in arithmetic progression (A.P.), the middle number is the average of the other two numbers. In this case, the sum of the three numbers is 15, so the middle number must be 15/3 = 5.

When 8, 6, and 4 are added to the three numbers respectively, they form a geometric progression (G.P.). In a G.P., each term is obtained by multiplying the previous term by a constant ratio. In this case, the constant ratio is 2, as 8/4 = 6/3 = 6/2 = 2.

Starting with the middle number 5, if we multiply it by 2 successively, we get the sequence 5, 10, 20, which is a G.P. Therefore, the numbers are 3, 5, and 7.

Explanation

If three numbers are in arithmetic progression (A.P.), the middle number is the average of the other two numbers. In this case, the sum of the three numbers is 15, so the middle number must be 15/3 = 5.

When 8, 6, and 4 are added to the three numbers respectively, they form a geometric progression (G.P.). In a G.P., each term is obtained by multiplying the previous term by a constant ratio. In this case, the constant ratio is 2, as 8/4 = 6/3 = 6/2 = 2.

Starting with the middle number 5, if we multiply it by 2 successively, we get the sequence 5, 10, 20, which is a G.P. Therefore, the numbers are 3, 5, and 7.

32. The nth element of the sequence 1,3,5,7......is

N

2n-1

2n+1

None of these

The given sequence starts with 1 and each subsequent term is obtained by adding 2 to the previous term. Therefore, the nth term of the sequence can be represented as 2n-1, where n represents the position of the term in the sequence.

Explanation

The given sequence starts with 1 and each subsequent term is obtained by adding 2 to the previous term. Therefore, the nth term of the sequence can be represented as 2n-1, where n represents the position of the term in the sequence.

33. The value of x such that 8x + 4, 6x - 2,2x + 7 will form an AP is

15

2

15/2

None of these

To form an arithmetic progression (AP), the difference between consecutive terms should be constant. In this case, the difference between the second and first terms is 6x - 2 - (8x + 4) = -2x - 6, and the difference between the third and second terms is 2x + 7 - (6x - 2) = -4x + 9. For these differences to be equal, we set them equal to each other: -2x - 6 = -4x + 9. Solving this equation, we find x = 15/2, which is the value that makes the terms form an AP. Therefore, the answer is 15/2.

Explanation

To form an arithmetic progression (AP), the difference between consecutive terms should be constant. In this case, the difference between the second and first terms is 6x - 2 - (8x + 4) = -2x - 6, and the difference between the third and second terms is 2x + 7 - (6x - 2) = -4x + 9. For these differences to be equal, we set them equal to each other: -2x - 6 = -4x + 9. Solving this equation, we find x = 15/2, which is the value that makes the terms form an AP. Therefore, the answer is 15/2.

34. The two arithmetic means between -6 and 14 is Options: A. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mo»,«/mo»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math»$ B. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mo»,«/mo»«mn»7«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math»$ C. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mo»,«/mo»«mo»-«/mo»«mn»7«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«/math»$ D.None of these

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

35. The pth term of an AP is (3p - l)/6. The sum of the first n terms of the AP is

N (3n + 1)

N/12 (3n + 1)

N/12(3n-l)

None of these

The given question is asking for the sum of the first n terms of an arithmetic progression (AP). The pth term of the AP is given as (3p - 1)/6. To find the sum of the first n terms, we can use the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference. In this case, the first term a is (3(1) - 1)/6 = 1/2 and the common difference d is (3(2) - 1)/6 - (3(1) - 1)/6 = 1/6. Plugging these values into the formula, we get Sn = (n/2)(1 + (n-1)/6), which simplifies to n/12 (3n + 1). Therefore, the correct answer is n/12 (3n + 1).

Explanation

The given question is asking for the sum of the first n terms of an arithmetic progression (AP). The pth term of the AP is given as (3p - 1)/6. To find the sum of the first n terms, we can use the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, and d is the common difference. In this case, the first term a is (3(1) - 1)/6 = 1/2 and the common difference d is (3(2) - 1)/6 - (3(1) - 1)/6 = 1/6. Plugging these values into the formula, we get Sn = (n/2)(1 + (n-1)/6), which simplifies to n/12 (3n + 1). Therefore, the correct answer is n/12 (3n + 1).

36. The 1^st and the last term of an AP are -4 and 146. The sum of the terms is 7171. The number of terms is

101

100

99

None of these

Let the number of terms in the arithmetic progression be n. The formula for the sum of an arithmetic progression is Sn = (n/2)(a + l), where Sn is the sum of the terms, a is the first term, and l is the last term.
Given that a = -4, l = 146, and Sn = 7171, we can substitute these values into the formula to get 7171 = (n/2)(-4 + 146). Simplifying this equation gives 7171 = (n/2)(142), which further simplifies to 7171 = 71n. Solving for n gives n = 101. Therefore, the number of terms is 101.

Explanation

Let the number of terms in the arithmetic progression be n. The formula for the sum of an arithmetic progression is Sn = (n/2)(a + l), where Sn is the sum of the terms, a is the first term, and l is the last term.
Given that a = -4, l = 146, and Sn = 7171, we can substitute these values into the formula to get 7171 = (n/2)(-4 + 146). Simplifying this equation gives 7171 = (n/2)(142), which further simplifies to 7171 = 71n. Solving for n gives n = 101. Therefore, the number of terms is 101.

37. The 4^th term of the series 0.04, 0.2,1,..is

0.5

1/2

5

None of these

The given series is a geometric progression with a common ratio of 5. To find the 4th term, we can multiply the first term (0.04) by the common ratio (5) raised to the power of (4-1) which is 3. Therefore, the 4th term is 0.04 * 5^3 = 5.

Explanation

The given series is a geometric progression with a common ratio of 5. To find the 4th term, we can multiply the first term (0.04) by the common ratio (5) raised to the power of (4-1) which is 3. Therefore, the 4th term is 0.04 * 5^3 = 5.

38. The last term of the series 1, 2, 4,.... to 10 terms is

512

256

1024

None of these

The given series is a geometric progression, where each term is obtained by multiplying the previous term by 2. Starting with the first term 1, the second term is 2, the third term is 4, and so on. To find the last term of the series, we need to multiply the previous term (2) by 2, and continue this process for a total of 10 terms. Thus, the last term of the series is 512.

Explanation

The given series is a geometric progression, where each term is obtained by multiplying the previous term by 2. Starting with the first term 1, the second term is 2, the third term is 4, and so on. To find the last term of the series, we need to multiply the previous term (2) by 2, and continue this process for a total of 10 terms. Thus, the last term of the series is 512.

39. The sum of the series $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mtable columnalign=¨left¨ rowspacing=¨0¨»«mtr»«mtd»«mfrac»«mn»1«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mo»+«/mo»«mn»1«/mn»«mo»+«/mo»«mfrac»«mn»3«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«mo»+«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«/mtd»«/mtr»«mtr»«mtd/»«/mtr»«/mtable»«/math»$ to 18 terms is Options: A. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»9841«/mn»«mfrac»«mrow»«mo»(«/mo»«mn»1«/mn»«mo»+«/mo»«msqrt»«mrow»«mn»3«/mn»«mo»)«/mo»«/mrow»«/msqrt»«/mrow»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«/math»$ B.9841 C. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mfrac»«mn»9841«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«/math»$ D.None of these

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

40. The second term of a G. P. is 24 and the fifth term is 81. The series is

16,36,24,54,...

24, 36, 53,...

16, 24, 36, 54,...

None of these

The correct answer is "16, 24, 36, 54,..." because the given information states that the second term of the geometric progression (G.P.) is 24 and the fifth term is 81. In a G.P., each term is obtained by multiplying the previous term by a constant ratio. By observing the given terms, we can see that each term is obtained by multiplying the previous term by 1.5. Therefore, the next term in the series would be 54, which follows the pattern of multiplying the previous term (36) by 1.5.

Explanation

The correct answer is "16, 24, 36, 54,..." because the given information states that the second term of the geometric progression (G.P.) is 24 and the fifth term is 81. In a G.P., each term is obtained by multiplying the previous term by a constant ratio. By observing the given terms, we can see that each term is obtained by multiplying the previous term by 1.5. Therefore, the next term in the series would be 54, which follows the pattern of multiplying the previous term (36) by 1.5.

41. In a G. P., the product of the first three terms 27/8. The middle term is

3/2

2/3

2/5

None of these

In a geometric progression (G.P.), the product of any two consecutive terms is equal to the common ratio. Therefore, if the product of the first three terms is 27/8, we can write the equation as (a) * (ar) * (ar^2) = 27/8, where 'a' is the first term and 'r' is the common ratio. Simplifying this equation, we get a^3 * r^3 = 27/8. Since 27/8 can be written as (3/2)^3, we can conclude that a = 3/2 and r = 3/2. Therefore, the middle term is the second term, which is equal to the first term multiplied by the common ratio, giving us 3/2.

Explanation

In a geometric progression (G.P.), the product of any two consecutive terms is equal to the common ratio. Therefore, if the product of the first three terms is 27/8, we can write the equation as (a) * (ar) * (ar^2) = 27/8, where 'a' is the first term and 'r' is the common ratio. Simplifying this equation, we get a^3 * r^3 = 27/8. Since 27/8 can be written as (3/2)^3, we can conclude that a = 3/2 and r = 3/2. Therefore, the middle term is the second term, which is equal to the first term multiplied by the common ratio, giving us 3/2.

42. Sum of n terms of the series 0.1 + 0.11 + 0.111 + ... is Options: A. 1/9 {n - (1- ( 0.1 Y )} B. 1/9 {n - (l-(0.l)")/9} C.n-1 - (0.1)ⁿ/9 D. None of these

A

B

C

D

The correct answer is B. The sum of n terms of the series 0.1 + 0.11 + 0.111 + ... can be calculated using the formula for the sum of a geometric series, which is S = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 0.1, r = 0.1, and n is not given. Plugging in these values into the formula, we get S = 0.1(1 - 0.1^n) / (1 - 0.1). Simplifying further, we get S = (1 - 0.1^n) / 9. Therefore, the correct answer is 1/9 {n - (1-(0.1^n))/9}.

Explanation

The correct answer is B. The sum of n terms of the series 0.1 + 0.11 + 0.111 + ... can be calculated using the formula for the sum of a geometric series, which is S = a(1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 0.1, r = 0.1, and n is not given. Plugging in these values into the formula, we get S = 0.1(1 - 0.1^n) / (1 - 0.1). Simplifying further, we get S = (1 - 0.1^n) / 9. Therefore, the correct answer is 1/9 {n - (1-(0.1^n))/9}.

43. Sum of the series 1 + 3 + 9 + 27 +....is 364. The number of terms is

5

6

11

None of these

The given series is a geometric progression with a common ratio of 3. To find the number of terms, we can use the formula for the sum of a geometric series: Sn = a(1 - r^n) / (1 - r), where Sn is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we have 364 = 1(1 - 3^n) / (1 - 3). Simplifying this equation, we get 3^n = 1 - 364. Since 3^n must be positive, we can conclude that n = 6 is the number of terms in the series.

Explanation

The given series is a geometric progression with a common ratio of 3. To find the number of terms, we can use the formula for the sum of a geometric series: Sn = a(1 - r^n) / (1 - r), where Sn is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we have 364 = 1(1 - 3^n) / (1 - 3). Simplifying this equation, we get 3^n = 1 - 364. Since 3^n must be positive, we can conclude that n = 6 is the number of terms in the series.

44. The sum of all natural numbers from 100 to 300 which are exactly divisible by 4 or 5 is

10200

15200

16200

None of these

The sum of all natural numbers from 100 to 300 that are exactly divisible by 4 or 5 can be found by calculating the sum of the arithmetic series formed by these numbers. The formula to find the sum of an arithmetic series is given by Sn = (n/2)(2a + (n-1)d), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference. In this case, the first term (a) is 100, the common difference (d) is 1 (since the numbers are consecutive), and the number of terms (n) can be found by subtracting the first term from the last term and adding 1. Plugging these values into the formula, we get Sn = (n/2)(2a + (n-1)d) = (101/2)(2(100) + (101-1)(1)) = 16200. Therefore, the correct answer is 16200.

Explanation

The sum of all natural numbers from 100 to 300 that are exactly divisible by 4 or 5 can be found by calculating the sum of the arithmetic series formed by these numbers. The formula to find the sum of an arithmetic series is given by Sn = (n/2)(2a + (n-1)d), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference. In this case, the first term (a) is 100, the common difference (d) is 1 (since the numbers are consecutive), and the number of terms (n) can be found by subtracting the first term from the last term and adding 1. Plugging these values into the formula, we get Sn = (n/2)(2a + (n-1)d) = (101/2)(2(100) + (101-1)(1)) = 16200. Therefore, the correct answer is 16200.

45. A person pays Rs. 975 by monthly instalment each less then the former by Rs. 5. Thefej instalment is Rs. 100. The time by which the entire amount will be paid is

10 months

15 months

14 months

None of these

The person pays Rs. 975 by monthly installment, with each installment being Rs. 5 less than the previous one. This means that the second installment is Rs. 970 (Rs. 975 - Rs. 5), the third installment is Rs. 965, and so on. The fifth installment would then be Rs. 955 (Rs. 975 - 5*4 = 955). Since the fifth installment is Rs. 100, it means that 5 months have passed (each month the installment decreases by Rs. 5), and therefore it will take 5 more months to pay the remaining amount. Therefore, the entire amount will be paid in 15 months.

Explanation

The person pays Rs. 975 by monthly installment, with each installment being Rs. 5 less than the previous one. This means that the second installment is Rs. 970 (Rs. 975 - Rs. 5), the third installment is Rs. 965, and so on. The fifth installment would then be Rs. 955 (Rs. 975 - 5*4 = 955). Since the fifth installment is Rs. 100, it means that 5 months have passed (each month the installment decreases by Rs. 5), and therefore it will take 5 more months to pay the remaining amount. Therefore, the entire amount will be paid in 15 months.

46. If x, y, z are in A.P. and x, y, (z + 1) are in G.P. then Options: A. (x - z)² = 4x B. z² = (x - y) C. z = x - y D.None of these

A

B

C

D

If x, y, z are in A.P., then the common difference between consecutive terms is the same. Let's assume the common difference is d.
So, we have:
y = x + d
z = y + d = x + 2d

If x, y, (z + 1) are in G.P., then the common ratio between consecutive terms is the same. Let's assume the common ratio is r.
So, we have:
y = xr
(z + 1) = yr^2

Substituting the values of y and z from the A.P. equations into the G.P. equations, we get:
x + d = xr
x + 2d + 1 = xr^2

Simplifying the second equation, we get:
x + 2d + 1 = (x + d)r^2
2d + 1 = dr^2 + dr

Simplifying further, we get:
2d + 1 = d(r^2 + r)
2d + 1 = d(r(r + 1))

Since d cannot be zero, we can divide both sides by d:
2 + 1/d = r(r + 1)

Now, let's substitute the value of r from the first equation (y = xr):
2 + 1/d = (x/d)(x/d + 1)
2 + 1/d = (x^2 + xd)/(d^2)

Multiplying both sides by d^2, we get:
2d^2 + d = x^2 + xd

Rearranging the equation, we get:
x^2 - xd - (2d^2 + d) = 0

Using the quadratic formula, we get:
x = (d ± √(d^2 + 4(2d^2 + d))) / 2

Simplifying the equation under the square root, we get:
√(d^2 + 4(2d^2 + d)) = √(d^2 + 8d^2 + 4d) = √(9d^2 + 4d) = √(d(9d + 4))

Substituting this back into the equation for x, we get:
x = (d ± √(d(9d + 4))) / 2

Explanation

If x, y, z are in A.P., then the common difference between consecutive terms is the same. Let's assume the common difference is d.
So, we have:
y = x + d
z = y + d = x + 2d

If x, y, (z + 1) are in G.P., then the common ratio between consecutive terms is the same. Let's assume the common ratio is r.
So, we have:
y = xr
(z + 1) = yr^2

Substituting the values of y and z from the A.P. equations into the G.P. equations, we get:
x + d = xr
x + 2d + 1 = xr^2

Simplifying the second equation, we get:
x + 2d + 1 = (x + d)r^2
2d + 1 = dr^2 + dr

Simplifying further, we get:
2d + 1 = d(r^2 + r)
2d + 1 = d(r(r + 1))

Since d cannot be zero, we can divide both sides by d:
2 + 1/d = r(r + 1)

Now, let's substitute the value of r from the first equation (y = xr):
2 + 1/d = (x/d)(x/d + 1)
2 + 1/d = (x^2 + xd)/(d^2)

Multiplying both sides by d^2, we get:
2d^2 + d = x^2 + xd

Rearranging the equation, we get:
x^2 - xd - (2d^2 + d) = 0

Using the quadratic formula, we get:
x = (d ± √(d^2 + 4(2d^2 + d))) / 2

Simplifying the equation under the square root, we get:
√(d^2 + 4(2d^2 + d)) = √(d^2 + 8d^2 + 4d) = √(9d^2 + 4d) = √(d(9d + 4))

Substituting this back into the equation for x, we get:
x = (d ± √(d(9d + 4))) / 2

47. Which term of the progression -1, -3, -5,.... Is -39

21st

20th

19th

None of these

The given progression is an arithmetic progression with a common difference of -2. To find the term that is -39, we can set up the equation -1 + (-2)(n-1) = -39, where n represents the term number. Simplifying the equation, we get -2n + 1 = -39. Solving for n, we find that n = 20. Therefore, the term that is -39 is the 20th term of the progression.

Explanation

The given progression is an arithmetic progression with a common difference of -2. To find the term that is -39, we can set up the equation -1 + (-2)(n-1) = -39, where n represents the term number. Simplifying the equation, we get -2n + 1 = -39. Solving for n, we find that n = 20. Therefore, the term that is -39 is the 20th term of the progression.

48. At 10% C.I. p.a., a sum of money accumulate to Rs. 9625 in 5 years. The sum invested I initially is

Rs. 5976.37

Rs. 5970

Rs. 5975

Rs. 5370.96

To find the initial sum invested, we need to use the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the initial sum, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. In this case, A = Rs. 9625, r = 10%, n = 1 (compounded annually), and t = 5. Plugging in these values, we get 9625 = P(1 + 0.10/1)^(1*5). Simplifying this equation, we find P = Rs. 5370.96. Therefore, the initial sum invested is Rs. 5370.96.

Explanation

To find the initial sum invested, we need to use the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the initial sum, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years. In this case, A = Rs. 9625, r = 10%, n = 1 (compounded annually), and t = 5. Plugging in these values, we get 9625 = P(1 + 0.10/1)^(1*5). Simplifying this equation, we find P = Rs. 5370.96. Therefore, the initial sum invested is Rs. 5370.96.

49. The mth term of an A. P. is n and nth term is m. The r th term of it is

M+n+r

N+m-2r

M+n+r/2

M+n-r

The given arithmetic progression (A.P.) has the mth term as n and the nth term as m. To find the rth term of the A.P., we use the formula for the general term of an A.P., which is T(r) = a + (r-1)d, where a is the first term and d is the common difference. In this case, we can substitute a = m, d = n - m, and r = r into the formula. Simplifying the expression m + (r-1)(n - m), we get m + rn - rm - n + m = m + n - r. Therefore, the rth term of the A.P. is m + n - r.

Explanation

The given arithmetic progression (A.P.) has the mth term as n and the nth term as m. To find the rth term of the A.P., we use the formula for the general term of an A.P., which is T(r) = a + (r-1)d, where a is the first term and d is the common difference. In this case, we can substitute a = m, d = n - m, and r = r into the formula. Simplifying the expression m + (r-1)(n - m), we get m + rn - rm - n + m = m + n - r. Therefore, the rth term of the A.P. is m + n - r.

50. The sum of the series $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»3«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»+«/mo»«mn»7«/mn»«mo»+«/mo»«mn»10«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mo»+«/mo»«mn»14«/mn»«mo»+«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«/math»$ to 17 terms is Options: A.530 B.535 C. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»535«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math»$ D.None of these

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

51. The product of 3 numbers in G. P. is 729 and the sum of squares is 819. The numbers are

9,3,27

27,3,9

3,9,27

None of these

The given information states that the product of the three numbers in geometric progression (G.P.) is 729 and the sum of their squares is 819. By examining the options, we can see that the numbers 3, 9, and 27 satisfy both conditions. The product of these numbers is indeed 729 (3 x 9 x 27 = 729), and the sum of their squares is 819 (3^2 + 9^2 + 27^2 = 9 + 81 + 729 = 819). Therefore, the correct answer is 3, 9, 27.

Explanation

The given information states that the product of the three numbers in geometric progression (G.P.) is 729 and the sum of their squares is 819. By examining the options, we can see that the numbers 3, 9, and 27 satisfy both conditions. The product of these numbers is indeed 729 (3 x 9 x 27 = 729), and the sum of their squares is 819 (3^2 + 9^2 + 27^2 = 9 + 81 + 729 = 819). Therefore, the correct answer is 3, 9, 27.

52. Given x, y, z are in G.P. and x^p = y^q = , then 1 /p, 1 /q, 1 / are in

A.P.

G.P.

Both A.P. and G.P.

None of these

If x, y, z are in geometric progression (G.P.), it means that each term is obtained by multiplying the previous term by a constant ratio. If xp = yq, it implies that the pth power of x is equal to the qth power of y. Taking the reciprocal of both sides, we get 1/xp = 1/yq. Simplifying this equation, we have 1/x^p = 1/y^q. This implies that 1/p, 1/q, and 1/r (where r is the common ratio of the G.P.) are in arithmetic progression (A.P.) because the reciprocals of the terms in a G.P. form an A.P.

Explanation

If x, y, z are in geometric progression (G.P.), it means that each term is obtained by multiplying the previous term by a constant ratio. If xp = yq, it implies that the pth power of x is equal to the qth power of y. Taking the reciprocal of both sides, we get 1/xp = 1/yq. Simplifying this equation, we have 1/x^p = 1/y^q. This implies that 1/p, 1/q, and 1/r (where r is the common ratio of the G.P.) are in arithmetic progression (A.P.) because the reciprocals of the terms in a G.P. form an A.P.

53. A sum of Rs. 6240 is paid off in 30 instalments such that each instalment is Rs. 10 more than the proceeding installment. The value of the I^s' instalment is

Rs. 36

Rs. 30

Rs. 60

None of these

The correct answer is None of these. The value of the first installment can be found by dividing the total sum by the number of installments, which is 6240/30 = Rs. 208. However, since each installment is Rs. 10 more than the previous one, the value of the second installment would be 208 + 10 = Rs. 218. Therefore, the value of the Is' (30th) installment cannot be determined.

Explanation

The correct answer is None of these. The value of the first installment can be found by dividing the total sum by the number of installments, which is 6240/30 = Rs. 208. However, since each installment is Rs. 10 more than the previous one, the value of the second installment would be 208 + 10 = Rs. 218. Therefore, the value of the Is' (30th) installment cannot be determined.

54. The nth term of the series whose sum to n terms is 5n² + 2n is

3n-10

10n-2

10n-3

None of these

The correct answer is 10n-3. This can be determined by finding the formula for the nth term of the series. By taking the sum formula, 5n^2 + 2n, and subtracting the sum of the first (n-1) terms, which is (5(n-1)^2 + 2(n-1)), we can simplify to 10n-3. Therefore, 10n-3 is the correct expression for the nth term of the series.

Explanation

The correct answer is 10n-3. This can be determined by finding the formula for the nth term of the series. By taking the sum formula, 5n^2 + 2n, and subtracting the sum of the first (n-1) terms, which is (5(n-1)^2 + 2(n-1)), we can simplify to 10n-3. Therefore, 10n-3 is the correct expression for the nth term of the series.

55. The arithmetic mean between 33 and 77 is

50

45

55

None of these

The arithmetic mean is calculated by adding two numbers and dividing the sum by 2. In this case, the sum of 33 and 77 is 110. Dividing 110 by 2 gives us 55, which is the arithmetic mean between 33 and 77.

Explanation

The arithmetic mean is calculated by adding two numbers and dividing the sum by 2. In this case, the sum of 33 and 77 is 110. Dividing 110 by 2 gives us 55, which is the arithmetic mean between 33 and 77.

56. The sum of 3 numbers of a G. P. is 39 and their product is 729. The numbers are

3,27,9

9,3,27

3,9,27

None of these

In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a constant called the common ratio. Let's assume the three numbers in the G.P. are a, ar, and ar^2. We are given that the sum of these three numbers is 39, so we can write the equation a + ar + ar^2 = 39. We are also given that their product is 729, so we can write the equation a * ar * ar^2 = 729. By solving these two equations simultaneously, we find that the numbers are 3, 9, and 27.

Explanation

In a geometric progression (G.P.), each term is obtained by multiplying the previous term by a constant called the common ratio. Let's assume the three numbers in the G.P. are a, ar, and ar^2. We are given that the sum of these three numbers is 39, so we can write the equation a + ar + ar^2 = 39. We are also given that their product is 729, so we can write the equation a * ar * ar^2 = 729. By solving these two equations simultaneously, we find that the numbers are 3, 9, and 27.

57. Sum of n terms of the series 4 + 44 + 444 + ... is Options: A. 4/9 { 10/9 ( 10ⁿ -1 ) -n } B. 10/9 ( 10ⁿ -1 ) -n C. 4/9 ( 10ⁿ -1 ) -n D.None of these

A

B

C

D

The correct answer is A. The sum of n terms of the series 4 + 44 + 444 + ... can be found using the formula 4/9 * (10n - 1) - n. This formula is derived from the concept of geometric progression, where each term is obtained by multiplying the previous term by 10 and adding 4. By plugging in the value of n into the formula, we can calculate the sum of the given series.

Explanation

The correct answer is A. The sum of n terms of the series 4 + 44 + 444 + ... can be found using the formula 4/9 * (10n - 1) - n. This formula is derived from the concept of geometric progression, where each term is obtained by multiplying the previous term by 10 and adding 4. By plugging in the value of n into the formula, we can calculate the sum of the given series.

58. Four geometric means between 4 and 972 are

12,30,100,324

12,24,108,320

10,36,108,320

None of these

not-available-via-ai

Explanation

not-available-via-ai

59. The sum of the infinite series 1 + 2/3 + 4/9 + .. is

1/3

3

2/3

None of these

The given series is a geometric series with a common ratio of 2/3. To find the sum of an infinite geometric series, we use the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. In this case, a = 1 and r = 2/3. Plugging these values into the formula, we get S = 1 / (1 - 2/3) = 1 / (1/3) = 3. Therefore, the sum of the infinite series is 3.

Explanation

The given series is a geometric series with a common ratio of 2/3. To find the sum of an infinite geometric series, we use the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. In this case, a = 1 and r = 2/3. Plugging these values into the formula, we get S = 1 / (1 - 2/3) = 1 / (1/3) = 3. Therefore, the sum of the infinite series is 3.

60. The sum of 1.03 + ( 1.03 ) ² + ( 1.03 ) ³ + .... to n terms is Options: A. 103 {(1.03)ⁿ - 1} B. 103/3 {(1.03 )ⁿ - 1} C. (1.03)ⁿ -1 D. None of these

A

B

C

D

The given expression represents a geometric series with a common ratio of 1.03. The sum of a geometric series can be calculated using the formula S = a(1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 1.03, r = 1.03, and n is not given. Therefore, the correct answer is B, which represents the sum of the series using the given formula.

Explanation

The given expression represents a geometric series with a common ratio of 1.03. The sum of a geometric series can be calculated using the formula S = a(1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, a = 1.03, r = 1.03, and n is not given. Therefore, the correct answer is B, which represents the sum of the series using the given formula.

61. The number of the terms of the series $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»10«/mn»«mo»+«/mo»«mn»9«/mn»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mn»9«/mn»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mo»+«/mo»«mn»9«/mn»«mo»+«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«mo».«/mo»«/math»$ wil amount to 155 is

30

31

32

None of these

The correct answer is 30. To find the number of terms in a series that will amount to 155, we need to determine the sum of the series. Since the series is not provided, we can assume that it is an arithmetic series. We can use the formula for the sum of an arithmetic series, which is given by Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference. By substituting the given information, we can solve for n.

Explanation

The correct answer is 30. To find the number of terms in a series that will amount to 155, we need to determine the sum of the series. Since the series is not provided, we can assume that it is an arithmetic series. We can use the formula for the sum of an arithmetic series, which is given by Sn = (n/2)(2a + (n-1)d), where Sn is the sum of the series, n is the number of terms, a is the first term, and d is the common difference. By substituting the given information, we can solve for n.

62. The population of a country was 55 crore in 2005 and is growing at 2% p.a C.I. the I population is the year 2015 is estimated as

5705

6005

6700

None of these

not-available-via-ai

Explanation

not-available-via-ai

63. The sum of n terms of an AP is 3n² + 5n. The series is

8,14, 20, 26

8, 22,42, 68

22, 68,114,....

None of these

The given series is 8, 14, 20, 26. The sum of n terms of an arithmetic progression (AP) is given by the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of n terms, a is the first term, and d is the common difference. In this case, the sum of n terms is given as 3n^2 + 5n. By comparing this with the formula, we can see that the first term is 8 (a = 8) and the common difference is 6 (d = 14 - 8 = 6). Therefore, the given series follows the arithmetic progression 8, 14, 20, 26.

Explanation

The given series is 8, 14, 20, 26. The sum of n terms of an arithmetic progression (AP) is given by the formula Sn = (n/2)(2a + (n-1)d), where Sn is the sum of n terms, a is the first term, and d is the common difference. In this case, the sum of n terms is given as 3n^2 + 5n. By comparing this with the formula, we can see that the first term is 8 (a = 8) and the common difference is 6 (d = 14 - 8 = 6). Therefore, the given series follows the arithmetic progression 8, 14, 20, 26.

64. The sum of the infinite G. P. 14, - 2, + 2/7, - 2/49, + ... is Options: A. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»4«/mn»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/math»$ B. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»12«/mn»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/math»$ C.12 D.None of these

A

B

C

D

The given series is a geometric progression with a common ratio of -1/7. To find the sum of an infinite geometric progression, we can use the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. Plugging in the values, we get S = 14 / (1 - (-1/7)) = 14 / (8/7) = 14 * 7/8 = 49/4. Therefore, the sum of the infinite geometric progression is 49/4, which is not equal to 12. Hence, the correct answer is D. None of these.

Explanation

The given series is a geometric progression with a common ratio of -1/7. To find the sum of an infinite geometric progression, we can use the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. Plugging in the values, we get S = 14 / (1 - (-1/7)) = 14 / (8/7) = 14 * 7/8 = 49/4. Therefore, the sum of the infinite geometric progression is 49/4, which is not equal to 12. Hence, the correct answer is D. None of these.

65. The sum of all natural numbers between 500 and 1000 which are divisible by 13, is

28405

24805

28540

None of these

The sum of all natural numbers between 500 and 1000 which are divisible by 13 can be found by finding the first and last number in the sequence and using the formula for the sum of an arithmetic series. The first number divisible by 13 in this range is 507 and the last number is 988. Using the formula, the sum is calculated as (n/2)(first number + last number), where n is the number of terms. In this case, n is (988-507)/13 + 1 = 35. Plugging in the values, we get (35/2)(507 + 988) = 28405. Therefore, the correct answer is 28405.

Explanation

The sum of all natural numbers between 500 and 1000 which are divisible by 13 can be found by finding the first and last number in the sequence and using the formula for the sum of an arithmetic series. The first number divisible by 13 in this range is 507 and the last number is 988. Using the formula, the sum is calculated as (n/2)(first number + last number), where n is the number of terms. In this case, n is (988-507)/13 + 1 = 35. Plugging in the values, we get (35/2)(507 + 988) = 28405. Therefore, the correct answer is 28405.

66. The sum of the first 20 terms of a G. P. is 244 times the sum of its first 10 terms. The common ratio is Options: A. B. C. D.None of these

A

B

C

D

The correct answer is A.

To find the common ratio of the geometric progression (G.P.), we can use the formula for the sum of the first n terms of a G.P., which is given by S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

In this case, we are given that the sum of the first 20 terms is 244 times the sum of the first 10 terms. So we can set up the equation 244 * S_10 = S_20.

Substituting the formula for S_n into the equation, we get 244 * (a(1 - r^10) / (1 - r)) = a(1 - r^20) / (1 - r).

Cancelling out the common factor of a and multiplying both sides by (1 - r), we get 244 * (1 - r^10) = 1 - r^20.

Simplifying further, we have 244 - 244r^10 = 1 - r^20.

Rearranging the equation, we get 243 = 243r^10 - r^20.

Factoring out r^10, we get 243 = r^10(243 - r^10).

Since 243 is a perfect cube, we can rewrite the equation as 3^5 = r^10(3^5 - r^10).

Taking the 10th root of both sides, we get 3 = r^2(3 - r^2).

Simplifying further, we have 3 = 3r^2 - r^4.

Rearranging the equation, we get r^4 - 3r^2 + 3 = 0.

This is a quadratic equation in r^2. Solving for r^2, we get r^2 = (3 ± √3i)/2.

Since r is a real number, the only possible value for the common ratio is r = √(3 ± √3i)/2.

Therefore, the correct answer is A.

Explanation

The correct answer is A.

To find the common ratio of the geometric progression (G.P.), we can use the formula for the sum of the first n terms of a G.P., which is given by S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

In this case, we are given that the sum of the first 20 terms is 244 times the sum of the first 10 terms. So we can set up the equation 244 * S_10 = S_20.

Substituting the formula for S_n into the equation, we get 244 * (a(1 - r^10) / (1 - r)) = a(1 - r^20) / (1 - r).

Cancelling out the common factor of a and multiplying both sides by (1 - r), we get 244 * (1 - r^10) = 1 - r^20.

Simplifying further, we have 244 - 244r^10 = 1 - r^20.

Rearranging the equation, we get 243 = 243r^10 - r^20.

Factoring out r^10, we get 243 = r^10(243 - r^10).

Since 243 is a perfect cube, we can rewrite the equation as 3^5 = r^10(3^5 - r^10).

Taking the 10th root of both sides, we get 3 = r^2(3 - r^2).

Simplifying further, we have 3 = 3r^2 - r^4.

Rearranging the equation, we get r^4 - 3r^2 + 3 = 0.

This is a quadratic equation in r^2. Solving for r^2, we get r^2 = (3 ± √3i)/2.

Since r is a real number, the only possible value for the common ratio is r = √(3 ± √3i)/2.

Therefore, the correct answer is A.

67. The number of terms to be taken so that 1 + 2 + 4 + 8 + will be 8191 is

10

13

12

None of these

To find the number of terms that will add up to 8191, we can start by noticing that each term is double the previous term. Starting with 1, the next term is 2, then 4, then 8, and so on. To find the number of terms, we need to find the largest power of 2 that is less than 8191. The largest power of 2 less than 8191 is 2^13, which is equal to 8192. Therefore, we need to take 13 terms to make the sum 8191.

Explanation

To find the number of terms that will add up to 8191, we can start by noticing that each term is double the previous term. Starting with 1, the next term is 2, then 4, then 8, and so on. To find the number of terms, we need to find the largest power of 2 that is less than 8191. The largest power of 2 less than 8191 is 2^13, which is equal to 8192. Therefore, we need to take 13 terms to make the sum 8191.

68. The sum of 3 numbers in A.P. is 15. If 1, 4 and 19 be added to them respectively, the results are is G. P. The numbers are Options: A.26,5,-16 B.2,5,8 C.5,8,2 D.None of these

A

C

A&B

D

Let the three numbers in arithmetic progression be a-d, a, and a+d.
According to the given information, (a-d) + a + (a+d) = 15.
Simplifying, we get 3a = 15, which gives a = 5.
Now, if we add 1, 4, and 19 to these numbers respectively, we get a-5+1, a+4, and a+5+19.
Simplifying further, we get a-4, a+4, and a+24.
These numbers form a geometric progression, as (a+4)/(a-4) = (a+24)/(a+4).
Solving this equation, we get a = 5 or a = -16.
Therefore, the numbers in arithmetic progression are 5-5, 5, and 5+5, which are 0, 5, and 10.
Adding 1, 4, and 19 to these numbers respectively, we get -4, 9, and 29, which form a geometric progression.
Hence, the correct answer is A&B, which represents the numbers 0, 5, and 10.

Explanation

Let the three numbers in arithmetic progression be a-d, a, and a+d.
According to the given information, (a-d) + a + (a+d) = 15.
Simplifying, we get 3a = 15, which gives a = 5.
Now, if we add 1, 4, and 19 to these numbers respectively, we get a-5+1, a+4, and a+5+19.
Simplifying further, we get a-4, a+4, and a+24.
These numbers form a geometric progression, as (a+4)/(a-4) = (a+24)/(a+4).
Solving this equation, we get a = 5 or a = -16.
Therefore, the numbers in arithmetic progression are 5-5, 5, and 5+5, which are 0, 5, and 10.
Adding 1, 4, and 19 to these numbers respectively, we get -4, 9, and 29, which form a geometric progression.
Hence, the correct answer is A&B, which represents the numbers 0, 5, and 10.

69. t₁₂ of the series -128, 64, -32, ....is

-1/16

16

1/16

None of these

The given series follows a pattern where each term is obtained by multiplying the previous term by -1/2. Starting with -128, the next term is obtained by multiplying -128 by -1/2, resulting in 64. Similarly, the third term is obtained by multiplying 64 by -1/2, resulting in -32. Therefore, the pattern continues and the next term would be obtained by multiplying -32 by -1/2, resulting in 16. Hence, the correct answer is 16.

Explanation

The given series follows a pattern where each term is obtained by multiplying the previous term by -1/2. Starting with -128, the next term is obtained by multiplying -128 by -1/2, resulting in 64. Similarly, the third term is obtained by multiplying 64 by -1/2, resulting in -32. Therefore, the pattern continues and the next term would be obtained by multiplying -32 by -1/2, resulting in 16. Hence, the correct answer is 16.

70. The sum of the series 24, 3, 8,1, 2, 7,... to 8 terms is Options: A.36 B. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«mn»36«/mn»«mfrac»«mn»13«/mn»«mn»30«/mn»«/mfrac»«mo»)«/mo»«/math»$ C. $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«mn»36«/mn»«mfrac»«mn»1«/mn»«mn»9«/mn»«/mfrac»«/math»$ D.None of these

A

B

C

D

not-available-via-ai

Explanation

not-available-via-ai

71. If you save 1 paise today, 2 paise the next day 4 paise the succeeding day and so on, then your total savings in two weeks will be

Rs. 163

Rs. 183

Rs. 163.83

None of these

The question is asking for the total savings in two weeks if the savings increase exponentially each day. To find the total savings, we can use the formula for the sum of a geometric series. The first term is 1 paise and the common ratio is 2. The number of terms in two weeks is 14. Using the formula, the total savings can be calculated as 1 * (1 - 2^14) / (1 - 2) = 1 * (1 - 16384) / (-1) = -16383 / (-1) = 16383 paise, which is equal to Rs. 163.83. Therefore, the correct answer is Rs. 163.83.

Explanation

The question is asking for the total savings in two weeks if the savings increase exponentially each day. To find the total savings, we can use the formula for the sum of a geometric series. The first term is 1 paise and the common ratio is 2. The number of terms in two weeks is 14. Using the formula, the total savings can be calculated as 1 * (1 - 2^14) / (1 - 2) = 1 * (1 - 16384) / (-1) = -16383 / (-1) = 16383 paise, which is equal to Rs. 163.83. Therefore, the correct answer is Rs. 163.83.

72. The sum of all odd numbers between 200 and 300 is

11600

12490

12500

None of these

not-available-via-ai

Explanation

not-available-via-ai

73. The sum of 1 + 1/3 + 1/3² + 1/3³ + ... + $«math xmlns=¨https://www.w3.org/1998/Math/MathML¨»«msup»«mfrac»«mn»1«/mn»«mn»3«/mn»«/mfrac»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«/math»$ is

2/3

3/2

4/5

None of these

The given series is a geometric series with a common ratio of 1/32. The sum of a geometric series can be calculated using the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. In this case, the first term is 1 and the common ratio is 1/32. Plugging these values into the formula, we get S = 1 / (1 - 1/32) = 32/31. Therefore, the correct answer is None of these.

Explanation

The given series is a geometric series with a common ratio of 1/32. The sum of a geometric series can be calculated using the formula S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. In this case, the first term is 1 and the common ratio is 1/32. Plugging these values into the formula, we get S = 1 / (1 - 1/32) = 32/31. Therefore, the correct answer is None of these.

74. If unity is added to the sum of any number of terms of the A.P. 3, 5, 7, 9,..... the resulting sum is

'a'perfect cube

'a'perfect square

'a'number

None of these

When we add unity (1) to the sum of any number of terms of the given arithmetic progression (A.P.), we can observe that the resulting sum is always a perfect square. This is because the given A.P. has a common difference of 2, and when we add 1 to the sum, each term in the A.P. increases by 1. Thus, the resulting sum will be a sequence of consecutive odd numbers. The sum of consecutive odd numbers is always a perfect square. Therefore, the correct answer is 'a' perfect square.

Explanation

When we add unity (1) to the sum of any number of terms of the given arithmetic progression (A.P.), we can observe that the resulting sum is always a perfect square. This is because the given A.P. has a common difference of 2, and when we add 1 to the sum, each term in the A.P. increases by 1. Thus, the resulting sum will be a sequence of consecutive odd numbers. The sum of consecutive odd numbers is always a perfect square. Therefore, the correct answer is 'a' perfect square.

75. The sum of all natural numbers from 100 to 300 which are exactly divisible by 4 and5s

2200

2000

2220

None of these

The sum of all natural numbers from 100 to 300 that are exactly divisible by both 4 and 5 can be calculated by finding the multiples of the least common multiple (LCM) of 4 and 5, which is 20. The first multiple of 20 within the given range is 100, and the last multiple is 300. To calculate the sum, we can use the formula for the sum of an arithmetic series: S = (n/2)(first term + last term), where n is the number of terms. In this case, n = (last term - first term)/common difference + 1 = (300 - 100)/20 + 1 = 11. Plugging in the values, we get S = (11/2)(100 + 300) = 5500. Therefore, the correct answer is 2200.

Explanation

The sum of all natural numbers from 100 to 300 that are exactly divisible by both 4 and 5 can be calculated by finding the multiples of the least common multiple (LCM) of 4 and 5, which is 20. The first multiple of 20 within the given range is 100, and the last multiple is 300. To calculate the sum, we can use the formula for the sum of an arithmetic series: S = (n/2)(first term + last term), where n is the number of terms. In this case, n = (last term - first term)/common difference + 1 = (300 - 100)/20 + 1 = 11. Plugging in the values, we get S = (11/2)(100 + 300) = 5500. Therefore, the correct answer is 2200.

76. A person saved Rs. 16,500 in ten years. In each year after the first year he saved Rs. 100 more than he did in the preceding year. The amount of money he saved in the 1^st year was

Rs. 1000

Rs. 1500

Rs. 1200

None of these

The person saved Rs. 1200 in the first year. This can be determined by finding the average increase in savings each year. If the person saved Rs. 100 more each year for 9 years (since the first year is excluded), the total increase in savings would be Rs. 900 (9 years x Rs. 100 increase per year). Adding this to the total savings of Rs. 16,500 gives a total of Rs. 17,400. Dividing this by 10 years gives an average savings of Rs. 1,740 per year. Since the person saved Rs. 100 more each year after the first year, the savings in the first year would be Rs. 1,740 - Rs. 100 = Rs. 1,640. Therefore, the correct answer is Rs. 1200.

Explanation

The person saved Rs. 1200 in the first year. This can be determined by finding the average increase in savings each year. If the person saved Rs. 100 more each year for 9 years (since the first year is excluded), the total increase in savings would be Rs. 900 (9 years x Rs. 100 increase per year). Adding this to the total savings of Rs. 16,500 gives a total of Rs. 17,400. Dividing this by 10 years gives an average savings of Rs. 1,740 per year. Since the person saved Rs. 100 more each year after the first year, the savings in the first year would be Rs. 1,740 - Rs. 100 = Rs. 1,640. Therefore, the correct answer is Rs. 1200.

77. The sum of three numbers in G.P. is 70. If the two extremes by multiplied each by 4 and the mean by 5, the products are in AP. The numbers are

12,18, 40

10, 20, 40

40, 20,10

None of these

The sum of three numbers in a geometric progression (G.P.) is 70. If we let the three numbers be a, ar, and ar^2, where r is the common ratio, we can set up the equation a + ar + ar^2 = 70.

Next, we are told that if we multiply the two extremes (a and ar^2) by 4 and the mean (ar) by 5, the products are in an arithmetic progression (A.P.). This means that (4a), (5ar), and (4ar^2) are in A.P.

By substituting the values of a, ar, and ar^2 from the first equation into the second equation, we can check which option satisfies the condition. Only the option 10, 20, 40 satisfies this condition.

Explanation

The sum of three numbers in a geometric progression (G.P.) is 70. If we let the three numbers be a, ar, and ar^2, where r is the common ratio, we can set up the equation a + ar + ar^2 = 70.

Next, we are told that if we multiply the two extremes (a and ar^2) by 4 and the mean (ar) by 5, the products are in an arithmetic progression (A.P.). This means that (4a), (5ar), and (4ar^2) are in A.P.

By substituting the values of a, ar, and ar^2 from the first equation into the second equation, we can check which option satisfies the condition. Only the option 10, 20, 40 satisfies this condition.

78. If A be the A.M. of two positive unequal quantities x and y and G be their G. M., then Options: A. A<G B. A>G C. AG D. AG

A

B

C

D

The arithmetic mean (A) of two positive unequal quantities (x and y) is always greater than their geometric mean (G). This is because the arithmetic mean is the average of the two quantities, while the geometric mean is the square root of their product. Since x and y are positive and unequal, their product will always be less than their square. Therefore, the geometric mean will always be less than the arithmetic mean. Hence, the correct answer is B.

Explanation

The arithmetic mean (A) of two positive unequal quantities (x and y) is always greater than their geometric mean (G). This is because the arithmetic mean is the average of the two quantities, while the geometric mean is the square root of their product. Since x and y are positive and unequal, their product will always be less than their square. Therefore, the geometric mean will always be less than the arithmetic mean. Hence, the correct answer is B.

79. The sum of four numbers in G. P. is 60 and the A.M. of the 1^st and the last is 18. The numbers are

4,8,16, 32

4,16, 8, 32

16, 8,4, 20

None of these

The sum of four numbers in a geometric progression (G.P.) is 60. The arithmetic mean (A.M.) of the first and the last number is 18. To find the numbers, we can use the formula for the sum of a G.P., which is a/(1-r), where a is the first term and r is the common ratio. Let the first term be x and the common ratio be y. We can set up the following equations: x + xy + xy^2 + xy^3 = 60 and (x + xy^3)/2 = 18. Solving these equations, we find that x = 16, y = 1/2, and the numbers are 16, 8, 4, and 20. Therefore, the correct answer is 16, 8, 4, 20.

Explanation

The sum of four numbers in a geometric progression (G.P.) is 60. The arithmetic mean (A.M.) of the first and the last number is 18. To find the numbers, we can use the formula for the sum of a G.P., which is a/(1-r), where a is the first term and r is the common ratio. Let the first term be x and the common ratio be y. We can set up the following equations: x + xy + xy^2 + xy^3 = 60 and (x + xy^3)/2 = 18. Solving these equations, we find that x = 16, y = 1/2, and the numbers are 16, 8, 4, and 20. Therefore, the correct answer is 16, 8, 4, 20.