Sequence And Series Arithmetic And Geometric Progressions

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1/79 Questions

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

About This Quiz

Explore the fundamentals of arithmetic and geometric progressions in this quiz. Focus on identifying nth elements, understanding sequence patterns, and solving related problems to enhance mathematical reasoning and problem-solving skills.

Sequence And Series Arithmetic And Geometric Progressions - Quiz

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2.

can be written as Options: A. B. C. D.None of these
- A
- B
- C
- D
Correct Answer

A. A

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.

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3.

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

A. 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.

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4.

The last term of the series 5, 7, 9,.... to 21 terms is
- 44
- 43
- 45
- None of these
Correct Answer

A. 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.

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5.

The number of numbers between 74 and 25556 divisible by 5 is
- 5090
- 5097
- 5095
- None of these
Correct Answer

A. 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.

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6.

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
Correct Answer

A. 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.

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7.

The last term of the series 1, -3, 9, -27 up to 7 terms is
- 297
- 729
- 927
- None of these
Correct Answer

A. 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.

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8.

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
Correct Answer

A. 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.

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9.

The sum of n terms of a G.P. whose first terms 1 and the common ratio is1/2, is equal to The value of n is
- 7
- 8
- 6
- None of these
Correct Answer

A. 8

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.

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10.

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
Correct Answer

A. C

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.

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11.

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
Correct Answer

A. 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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. (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).

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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
Correct Answer

A. -1,0,3

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.

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15.

The 20^th term of the progression 1, 4, 7,10.............. is
- 58
- 52
- 50
- None of these
Correct Answer

A. 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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. 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.

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19.

The sum of the series 9, 5, 1,.... to 100 terms is
- -18900
- 18900
- 19900
- None of these
Correct Answer

A. -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.

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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
Correct Answer

A. 2,5,8

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.

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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. B.6 C.4/11 None of these
- A
- B
- C
- D
Correct Answer

A. A

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.

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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
Correct Answer

A. B&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.

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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
Correct Answer

A. 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.

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24.

T₈ of the series 6,12, 24,.. .is
- 786
- 768
- 867
- None of these
Correct Answer

A. 768

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.

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25.

The sum of the infinite G. P. 1 - 1/3 + 1/9 - 1/27 +... is
- 0.33
- 0.57
- 0.75
- None of these
Correct Answer

A. 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.

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26.

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
Correct Answer

A. (16,4)

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.

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27.

The nth term of the series 16, 8, 4,.... Is 1 /2¹⁷. The value of n is
- 20
- 21
- 22
- None of these
Correct Answer

A. 22

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.

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28.

The nth element of the sequence 1,3,5,7......is
- N
- 2n-1
- 2n+1
- None of these
Correct Answer

A. 2n-1

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.

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29.

The 4 arithmetic means between -2 and 23 are
- 3,13,8,18
- 18, 3, 8,13
- 3, 8,13,18
- None of these
Correct Answer

A. 3, 8,13,18

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.

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30.

The 7^th term of the series 6, 12, 24,....... is
- 384
- 834
- 438
- None of these
Correct Answer

A. 384

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.

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31.

The sum of the series -2, 6, -18, .... to 7 terms is
- -1094
- 1094
- -1049
- None of these
Correct Answer

A. -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.

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32.

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
Correct Answer

A. 3, 5, 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.

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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
Correct Answer

A. 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.

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34.

The two arithmetic means between -6 and 14 is Options: A. B. C. D.None of these
- A
- B
- C
- D
Correct Answer

A. B
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
Correct Answer

A. 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).

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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
Correct Answer

A. 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.

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37.

The 4^th term of the series 0.04, 0.2,1,..is
- 0.5
- 1/2
- 5
- None of these
Correct Answer

A. 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.

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38.

The last term of the series 1, 2, 4,.... to 10 terms is
- 512
- 256
- 1024
- None of these
Correct Answer

A. 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.

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39.

The sum of the series to 18 terms is Options: A. B.9841 C. D.None of these
- A
- B
- C
- D
Correct Answer

A. A
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
Correct Answer

A. 16, 24, 36, 54,...

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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. B

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}.

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43.

Sum of the series 1 + 3 + 9 + 27 +....is 364. The number of terms is
- 5
- 6
- 11
- None of these
Correct Answer

A. 6

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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. A

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

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47.

Which term of the progression -1, -3, -5,.... Is -39
- 21st
- 20th
- 19th
- None of these
Correct Answer

A. 20th

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.

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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
Correct Answer

A. 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.

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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
Correct Answer

A. 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.

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