CBSE Class 10 Mathematics Arithmetic Progression Worksheet Set K

A. Very Short Answer Type Questions

Question. Write the first four terms of each of the following sequences whose \( n^{th} \) terms are -
(i) \( a_n = 3n + 2 \)
(ii) \( a_n = \frac{n - 2}{3} \)
(iii) \( a_n = 3^n \)
(iv) \( a_n = \frac{3n - 2}{5} \)
(v) \( a_n = (-1)^n \cdot 2^n \)
(vi) \( a_n = \frac{n(n - 2)}{2} \)
(vii) \( a_n = n^2 - n + 1 \)
(viii) \( a_n = 2n^2 - 3n + 1 \)
(ix) \( a_n = \frac{2n - 3}{6} \)
Answer: (i) 5, 8, 11, 14 (ii) \( -\frac{1}{3}, 0, \frac{1}{3}, \frac{2}{3} \) (iii) 3, 9, 27, 81 (iv) \( \frac{1}{5}, \frac{4}{5}, \frac{7}{5}, 2 \) (v) –2, 4, –8, 16 (vi) \( -\frac{1}{2}, 0, \frac{3}{2}, 4 \) (vii) 1, 3, 7, 13 (viii) 0, 3, 10, 21 (ix) \( -\frac{1}{6}, \frac{1}{6}, \frac{1}{2}, \frac{5}{6} \)

Question. The general term of a sequence is given by \( a_n = -4n + 15 \). Is the sequence an A.P. ? If so, find its \( 15^{th} \) term and the common difference.
Answer: –45, – 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 :
(i) \( 10^{th} \) term of the A.P. 1, 4, 7, 10, ....
(ii) \( 18^{th} \) term of the A.P. \( \sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, \dots \)
(iii) \( n^{th} \) term of the A.P. 13, 8, 3, –2, ....
Answer: (i) 28 (ii) \( 35\sqrt{2} \) (iii) \( -5n + 18 \)

Question. Which term of the A.P. 3, 8, 13, .... is 248 ?
Which term of the A.P. 84, 80, 76, .... is 0 ?
Which term of the A.P. 4, 9, 14, .... is 254 ?
Answer: (i) 50 (ii) 22 (iii) 51

Question. Is 68 a term of the A.P. 7, 10, 13, .... ?
Is 302 a term of the A.P. 3, 8, 13, .... ?
Answer: (i) No (ii) No

Question. How many terms are there in the A.P. 7, 10, 13, .... 43 ?
How many terms are there in the A.P. \( -1, -\frac{5}{6}, -\frac{2}{3}, -\frac{1}{2}, \dots, \frac{10}{3} \)?
Answer: (i) 13 (ii) 27

Question. The \( 10^{th} \) and \( 18^{th} \) terms of an A.P. are 41 and 73 respectively. Find \( 26^{th} \) term.
Answer: 105

Question. If 10 times the \( 10^{th} \) term of an A.P. is equal to 15 times the \( 15^{th} \) term, show that \( 25^{th} \) term of the A.P. is zero.
Answer: (Show that...)

Question. The \( 6^{th} \) and \( 17^{th} \) terms of an A.P. are 19 and 41 respectively, find the \( 40^{th} \) 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 \( 24^{th} \) term is twice the \( 10^{th} \) term. Prove that the \( 72^{nd} \) term is twice the \( 34^{th} \) term.
Answer: (Show that...)

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: (Show that...)

Question. If the \( n^{th} \) term of the A.P. 9, 7, 5, .... is same as the \( n^{th} \) 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 progressions :
(i) \( a + b, a - b, a - 3b, \dots \text{ to 22 terms} \)
(ii) \( (x - y)^2, (x^2 + y^2), (x + y)^2, \dots \text{ to } n \text{ terms} \)
(iii) \( \frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}, \dots \text{ to } n \text{ terms} \)
Answer: (i) 22a – 440b (ii) \( n[(x - y)^2 + (n - 1) xy] \) (iii) \( \frac{n}{2(x + y)} [n (2x - y) - y] \)

Question. Find the sum of \( n \) terms of an A.P. whose \( n^{th} \) 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 \( 12^{th} \) 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: (Show that...)

Question. In an A.P., if the \( 5^{th} \) and \( 12^{th} \) 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 \( 6^{th} \) years & 11300 in \( 9^{th} \) year find the production in (i) first year (ii) \( 8^{th} \) year (iii) \( 6^{th} \) year.
Answer: (i) 2500 (ii) 10200 (iii) 31500 (Wait, (iii) asks for 6th year which is already 8000. Answer Key for (iii) says 31500 which is sum of first six years).

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

EXERCISE # 2

Question. How many two digit number are there which are divisible by 7 ?
(a) 13
(b) 14
(c) 15
(d) None
Answer: (a) 13

Question. How many numbers are there between 103 and 750 which are divisible by 6 ?
(a) 125
(b) 108
(c) 107
(d) 113
Answer: (b) 108

Question. The sum of first 60 natural numbers is –
(a) 1830
(b) 1640
(c) 3660
(d) 1770
Answer: (a) 1830

Question. The sum of all 2 digit numbers is –
(a) 4750
(b) 4905
(c) 3776
(d) 4680
Answer: (b) 4905

Question. \( 23^{rd} \) term of the A.P. 7, 5, 3, 1, ....... is –
(a) 51
(b) 37
(c) –37
(d) –51
Answer: (c) –37

Question. If \( (k + 1) \), \( 3k \) and \( (4k + 2) \) be any three consecutive terms of an A.P., then the value of \( k \) is –
(a) 3
(b) 0
(c) 1
(d) 2
Answer: (a) 3

Question. Which term of the A.P. 5, 8, 11, 24 .... is 320 ?
(a) \( 104^{th} \)
(b) \( 105^{th} \)
(c) \( 106^{th} \)
(d) \( 64^{th} \)
Answer: (c) \( 106^{th} \)

Question. The \( 5^{th} \) and \( 13^{th} \) terms of an A.P. are 5 and – 3 respectively. The first term of the A.P. is –
(a) 1
(b) 9
(c) –15
(d) 2
Answer: (b) 9

Question. Which term of the A.P. 64, 60, 56, 52, .....is zero?
(a) \( 16^{th} \)
(b) \( 17^{th} \)
(c) \( 14^{th} \)
(d) \( 15^{th} \)
Answer: (b) \( 17^{th} \)

Question. The \( n^{th} \) term of an A.P. is \( (3n + 5) \). Its \( 7^{th} \) term is –
(a) 26
(b) \( (3n-2) \)
(c) \( 3n + 12 \)
(d) cannot be determined
Answer: (a) 26

Question. The sides of a right angle triangle are in A.P. The ratio of side is –
(a) 1 : 2 : 3
(b) 2 : 3 : 4
(c) 3 : 4 : 5
(d) 5 : 8 : 3
Answer: (c) 3 : 4 : 5

Question. The sum of 1, 3, 5, 7, 9, ....... upto 20 terms is–
(a) 400
(b) 563
(c) 472
(d) 264
Answer: (a) 400

Question. The sum of the series 5 + 13 + 21 + ... + 181 is –
(a) 2139
(b) 2476
(c) 2219
(d) 2337
Answer: (a) 2139

Question. The sum of all odd numbers between 100 and 200 is –
(a) 6200
(b) 6500
(c) 7500
(d) 3750
Answer: (c) 7500

Question. The sum of all positive integral multiples of 5 less than 100 is –
(a) 950
(b) 1230
(c) 760
(d) 875
Answer: (a) 950

Question. The sum of all even natural numbers less than 100 is –
(a) 2450
(b) 2272
(c) 2352
(d) 2468
Answer: (a) 2450

Question. Arithmetic mean between 14 and 18 is –
(a) 16
(b) 15
(c) 17
(d) 32
Answer: (a) 16

Question. If 4, \( A_1 \), \( A_2 \), \( A_3 \), 28 are in A.P., then the value of \( A_3 \) is –
(a) 23
(b) 22
(c) 19
(d) cannot be determined
Answer: (b) 22

Question. How many terms of the A.P. 3, 6, 9, 12, 15, ..... must be taken to make the sum 108 ?
(a) 6
(b) 7
(c) 8
(d) 36
Answer: (c) 8

Question. The \( 6^{th} \) and \( 8^{th} \) terms of an A.P. are 12 and 22 respectively, Its \( 2^{nd} \) term is –
(a) 8
(b) –8
(c) 6
(d) –3
Answer: (b) –8

Question. In an AP, then sum of first \( n \) terms is \( \left( \frac{3n^2}{2} + \frac{5n}{2} \right) \). Find its \( 25^{th} \) term.
(a) 924
(b) 76
(c) 1924
(d) 1848
Answer: (b) 76

Question. 200 logs are stocked in such a way that there are 20 logs in the bottom row, 19 in the next row, 18 in the next row and so on. In how many row 200 logs are placed and how many logs are there in the top row ?
(a) 19, 5
(b) 16, 5
(c) 10, 20
(d) 20, 7
Answer: (b) 16, 5