Sequence And Series Arithmetic And Geometric Progressions

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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}.

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.

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.

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.

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.

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.

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.

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.

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.