CBSE Class 10 Mathematics Arithmetic Progressions Worksheet Set 11

A. Very Short Answer Type Questions

Question. Write the first four terms of the sequence whose nth term is \( a_n = 3n + 2 \)
Answer: 5, 8, 11, 14

Question. Write the first four terms of the sequence whose nth term is \( a_n = \frac{n - 2}{3} \)
Answer: \( -\frac{1}{3}, 0, \frac{1}{3}, \frac{2}{3} \)

Question. Write the first four terms of the sequence whose nth term is \( a_n = 3^n \)
Answer: 3, 9, 27, 81

Question. Write the first four terms of the sequence whose nth term is \( a_n = \frac{3n - 2}{5} \)
Answer: \( \frac{1}{5}, \frac{3}{5}, \frac{7}{5}, 2 \)

Question. Write the first four terms of the sequence whose nth term is \( a_n = (-1)^n \cdot 2n \)
Answer: –2, 4, –8, 16

Question. Write the first four terms of the sequence whose nth term is \( a_n = \frac{n(n - 2)}{2} \)
Answer: \( -\frac{1}{2}, 0, \frac{3}{2}, 4 \)

Question. Write the first four terms of the sequence whose nth term is \( a_n = n^2 – n + 1 \)
Answer: 1, 3, 7, 13

Question. Write the first four terms of the sequence whose nth term is \( a_n = 2n^2 – 3n + 1 \)
Answer: 0, 3, 10, 21

Question. Write the first four terms of the sequence whose nth term is \( a_n = \frac{2n - 3}{6} \)
Answer: \( -\frac{1}{6}, \frac{1}{6}, \frac{1}{2}, \frac{5}{6} \)

Question. The general term of a sequence is given by \( a_n = – 4n + 15 \). Is the sequence an A.P. ? If so, find its 15th term and the common difference.
Answer: Yes, it is an A.P. 15th term is –45 and common difference is –4.

Question. The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
Answer: 26

B. Short answer type Questions

Question. Find the 10th term of the A.P. 1, 4, 7, 10, ....
Answer: 28

Question. Find the 18th term of the A.P. \( \sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, .... \)
Answer: \( 35\sqrt{2} \)

Question. Find the nth term of the A.P. 13, 8, 3, –2, ....
Answer: \( – 5n + 18 \)

Question. Which term of the A.P. 3, 8, 13, .... is 248 ?
Answer: 50

Question. Which term of the A.P. 84, 80, 76, .... is 0 ?
Answer: 22

Question. Which term of the A.P. 4, 9, 14, .... is 254 ?
Answer: 51

Question. Is 68 a term of the A.P. 7, 10, 13, .... ?
Answer: No

Question. Is 302 a term of the A.P. 3, 8, 13, .... ?
Answer: No

Question. How many terms are there in the A.P. 7, 10, 13, .... 43 ?
Answer: 13

Question. How many terms are there in the A.P. \( –1, –\frac{5}{6}, –\frac{2}{3}, –\frac{1}{2}, .... , \frac{10}{3} \) ?
Answer: 27

Question. The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.
Answer: 105

Question. If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
Answer: As \( 10(a + 9d) = 15(a + 14d) \), simplifying gives \( a + 24d = 0 \), hence the 25th term is zero.

Question. The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.
Answer: 87

Question. Find the sum of all odd numbers between 100 and 200.
Answer: 7500

Question. Find the sum of all integers between 84 and 719, which are multiples of 5.
Answer: 50800

Question. Find the sum of all integers between 50 and 500 which are divisible by 7.
Answer: 17696

C. Long answer type Questions

Question. In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.
Answer: Proof: Given \( a_{24} = 2a_{10} \), then \( a + 23d = 2(a + 9d) \implies a = 5d \). Now \( a_{72} = a + 71d = 5d + 71d = 76d \) and \( 2a_{34} = 2(a + 33d) = 2(5d + 33d) = 76d \). Hence proved.

Question. If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.
Answer: Proof follows using standard A.P. formula \( a_k = a + (k-1)d \).

Question. If the nth term of the A.P. 9, 7, 5, .... is same as the nth term of the A.P. 15, 12, 9, .... find n.
Answer: 7

Question. The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
Answer: 1, 7, 13

Question. Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.
Answer: 6, 9, 12

Question. The angles of a quadrilateral are in A.P. whose common difference is 10º. Find the angles.
Answer: 75º, 85º, 95º, 105º

Question. Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.
Answer: 5, 10, 15, 20

Question. Find the sum of the following arithmetic progression: a + b, a – b, a – 3b, .... to 22 terms
Answer: 22a – 440b

Question. Find the sum of the following arithmetic progression: \( (x – y)^2, (x^2 + y^2), (x + y)^2, .... \) to n terms
Answer: \( n[(x – y)^2 + (n – 1) xy] \)

Question. Find the sum of the following arithmetic progression: \( \frac{x-y}{x+y}, \frac{3x-2y}{x+y}, \frac{5x-3y}{x+y}, .... \) to n terms
Answer: \( \frac{n}{2(x + y)}[n (2x – y) – y] \)

Question. Find the sum of n terms of an A.P. whose nth terms is given by \( a_n = 5 – 6n \).
Answer: n (2 – 3n)

Question. How many terms are there in the A.P. whose first and fifth terms are – 14 and 2 respectively and the sum of the terms is 40 ?
Answer: 10

Question. The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
Answer: –1, 4, 740

Question. The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.
Answer: 3

Question. If 12th term of an A.P. is –13 and the sum of the first four terms is 24, what is the sum of first 10 terms ?
Answer: 0

Question. Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.
Answer: The numbers are 3, 9, 15... 999. Summing these terms using \( S_n \) formula confirms the value.

Question. In an A.P., if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms.
Answer: 1150

Question. The production of TV in a factory increases uniformly by a fixed number every year if produced 8000 acts in 6th years & 11300 in 9th year find the production in (i) first year (ii) 8th year (iii) 6th year.
Answer: (i) 2500 (ii) 10200 (iii) 31500

Question. A sum of Rs. 2800 is to be used to award four prizes. If each prize after the first prize is Rs. 200 less than the preceding prize, find the value of each of the prizes.
Answer: Rs. 1000, Rs. 800, Rs. 600, Rs. 400.