CBSE Class 10 Mathematics Arithmetic Progressions VBQs Set 09

Answer the following:

Question. Find the sum of first 10 terms of the AP: 2, 7, 12, ...
Answer: 245

Question. If the sum of first \( m \) terms of an AP is \( 2m^2 + 3m \), then what is its second term?
Answer: 9

Question. Find the sum of first 10 multiples of 6.
Answer: 330

Question. What is the sum of five positive integers divisible by 6?
Answer: 90

Question. If the sum of the first \( q \) terms of an AP is \( 2q + 3q^2 \), what is its common difference?
Answer: 6

Question. If \( n\text{th} \) term of an AP is \( (2n + 1) \), what is the sum of its first three terms?
Answer: 15

II. Short Answer Type Questions-I 

Question. Find the sum of first 8 multiples of 3.
Answer: 108

Question. Find the number of terms of the AP: 54, 51, 48, ... so that their sum is 513.
Answer: 18 or 19

Question. In an AP, the first term is –4, the last term is 29 and the sum of all its terms is 150. Find its common difference.
Answer: \( d = 3 \)

Question. Find the sum of all three digit natural numbers, which are multiples of 11.
Answer: 44550

Question. The first and the last terms of an AP are 8 and 65 respectively. If sum of all its terms is 730, find its common difference.
Answer: \( d = 3 \)

Question. The sum of the first \( n \) terms of an AP is \( 4n^2 + 2n \). Find the \( n\text{th} \) term of this AP.
Answer: \( a_n = 8n - 2 \)

Question. How many terms of the AP: 18, 16, 14, ... be taken so that their sum is zero?
Answer: 19

Question. In an AP, if \( S_5 + S_7 = 167 \) and \( S_{10} = 235 \), then find the AP, where \( S_n \) denotes the sum of its first \( n \) terms.
Answer: AP is 1, 6, 11, ...

Question. The sum of first \( n \) terms of an AP is given by \( S_n = 2n^2 + 3n \). Find the sixteenth term of the AP.
Answer: 65

III. Short Answer Type Questions-II 

Question. How many multiples of 4 lie between 10 and 250? Also find their sum.
Answer: 60 multiples; sum = 7800

Question. Find the sum of first \( n \) terms of an AP whose \( n\text{th} \) term is \( 5n – 1 \). Hence find the sum of first 20 terms.
Answer: \( S_n = \frac{n(5n + 3)}{2} \); \( S_{20} = 1030 \)

Question. The sum of first six terms of an AP is 42. The ratio of its 10th term to its 30th term is 1 : 3. Calculate the first and the thirteenth terms of the AP.
Answer: \( a = 2 \), \( a_{13} = 26 \)

Question. Find the sum of all multiples of 7 lying between 500 and 900.
Answer: \( n = 57 \), sum = 39,900

Question. If \( M \), \( N \) and \( T \) are in AP, prove that \( (M + 2N – T) (2N + T – M) (T + M – N) = 4MNT \).
Answer: As \( M \), \( N \), \( T \) are in AP, \( 2N = M + T \).
Substituting \( 2N \) in the expression:
\( (M + M + T - T)(M + T + T - M)(T + M - N) \)
\( = (2M)(2T)(M + T - N) \)
Since \( M + T = 2N \):
\( = (4MT)(2N - N) = 4MNT \).

Question. In an AP, if the 6th and 13th terms are 35 and 70 respectively, find the sum of its first 20 terms.
Answer: 1150

Question. The sum of the \( 2\text{nd} \) and the \( 7\text{th} \) terms of an AP is 30. If its 15th term is 1 less than twice its \( 8\text{th} \) term, find the AP.
Answer: AP is 1, 5, 9, 13, 17, ...

Question. If the ratio of the sum of first \( n \) terms of two AP’s is \( (7n + 1) : (4n + 27) \), find the ratio of their \( m\text{th} \) terms.
Answer: \( (14m - 6) : (8m + 23) \)

Question. The digits of a positive number of three digits are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number.
Answer: 852

Question. The sums of first \( n \) terms of three A.Ps’ are \( S_1, S_2 \) and \( S_3 \). The first term of each AP is 5 and their common differences are 2, 4 and 6 respectively. Prove that \( S_1 + S_3 = 2S_2 \).
Answer: \( S_1 = \frac{n}{2}[10 + (n-1)2] = 4n + n^2 \)
\( S_2 = \frac{n}{2}[10 + (n-1)4] = 3n + 2n^2 \)
\( S_3 = \frac{n}{2}[10 + (n-1)6] = 2n + 3n^2 \)
\( S_1 + S_3 = (4n + n^2) + (2n + 3n^2) = 6n + 4n^2 = 2(3n + 2n^2) = 2S_2 \).

IV. Long Answer Type Questions

Question. The sum of the first three numbers in an arithmetic progression is 18. If the product of the first and the third terms is 5 times the common difference, find the three numbers.
Answer: 2, 6, 10

Question. The first and the last term of an AP are 8 and 350 respectively. If its common difference is 9, how many terms are there and what is their sum?
Answer: \( n = 39 \); sum = 6981

Question. The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the \( 5\text{th} \) term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.
Answer: 1 : 3 and 5 : 49

Question. The sum of the first five terms of an AP is 55 and sum of the first ten terms of this AP is 235, find the sum of its first 20 terms.
Answer: 970

Question. The sums of \( n \) terms of two APs are in the ratio \( 5n + 4 : 9n + 6 \). Find the ratio of their 25th terms.
Answer: 249 : 447

Question. Find the middle term of the sequence formed by all three-digit numbers which leave a remainder 3, when divided by 4. Also, find the sum of all numbers on both sides of the middle terms separately.
Answer: Middle term = 551; sum of terms before = 36400; sum of terms after = 87024

Question. If the ratio of the sum of the first \( n \) terms of two APs is \( (7n + 1) : (4n + 27) \), then find the ratio of their 9th terms.
Answer: 24/19

Question. If the sum of first 14 terms of an AP is 1050 and its first term is 10, find the 20th term.
Answer: 200

Question. The first term of an AP is 5, the last term is 45 and sum is 400. Find the number of terms and the common difference.
Answer: \( n = 16 \); \( d = \frac{8}{3} \)

Question. How many terms of the AP: 24, 21, 18, ... must be taken so that their sum is 78?
Answer: 12