Worksheet on Chapter: Arithmetic Progressions - Class 10

Solved Questions on Arithmetic Progression

1. Which term of the A.P. 5, 14, 23, 32, ……. will be 198 more than its 32nd term?

a) 52^nd
b) 55^th
c) 54^th
d) 56^th

Answer: c) 54^th

Explanation: Given A.P.: 5, 14, 23, 32, …….

Thus,

a = 5
d = 14 − 5 = 9

We know that a_n = a + (n − 1) d

a₃₂ = 5 + (32 − 1) 9
= 5 + (31) 9
= 5 + 279
= 284

Let a_k be 198 more than the 32^nd term of the A.P.

a_k = a₃₂ + 198
= 284 + 198
= 482

Now we will find the value of k using an = a + (n − 1) d

a_k = a + (k − 1) d
482 = 5 + (k − 1) 9
482 − 5 = (k − 1) 9
477 = (k − 1) 9
477 / 9 = k − 1
53 = k − 1
53 + 1 = k
k = 54

Thus, 54^th term of the A.P. is 198 more than its 32^nd term.

2. What is the sum of the first 55 terms of the A.P. 13, 24, 35, ………..?

a) 17150
b) 17050
c) 17500
d) 17250

Answer: b) 17050

Explanation: Given A.P.: 13, 24, 35, ……….

Thus,

a = 13
d = 24 − 13 = 11
n = 55

We know that S_n = n ÷ 2 [2a + (n - 1) d]

S₅₅ = 55 ÷ 2 [2(13) + (55 - 1) 11]
= 55 ÷ 2 [26 + (54) 11]
= 55 ÷ 2 [26 + 594]
= 55 ÷ 2 (620)
= 55 × 310
= 17050

3. If the fourth term of an A.P. is 28 and the ninth term is 53, then what is the sum of the first 19 terms of the A.P.?

a) 1102
b) 1112
c) 1108
d) 1122

Answer: a) 1102

Explanation: Given: a₄ = 28 and a₉ = 53

We know that a_n = a + (n − 1) d

a₄ = a + 3d
28 = a + 3d … (1)
a₉ = a + 8d
53 = a + 8d … (2)

Now, subtracting (1) from (2)

53 − 28 = (a + 8d) − (a + 3d)
25 = 8d − 3d
25 = 5d
d = 25 / 5
d = 5

Putting the value of d = 5 in equation (1) to find the value of a.

28 = a + 3(5)
28 = a + 15
28 − 15 = a
a = 13

Now, we know that the sum of n terms of an A.P. is given by:

S_n = n ÷ 2[2a + (n - 1) d]

Here, n = 19

S₁₉ = 19 ÷ 2 [2(13) + (19 - 1)5]
= 19 ÷ 2 [26 + (18)5]
= 19 ÷ 2 [26 +90]
= (19 × 116) ÷ 2
= 19 × 58
= 1102

4. If the sum of three numbers in A.P. is 78 and their product is 16302, then what is the value of the common difference?

a) −7
b) −17
c) 7
d) 17

Answer: c) 7

Explanation: Let a − d, a, a + d be the three numbers in A.P.

We are given that their sum is 78.

(a − d) + a + (a + d) = 78
3a = 78
a = 78 / 3
a = 26

We are given that the product of the numbers is 16302.

(a − d) × a × (a + d) = 16302
(26 − d) × 26 × (26 + d) = 16302
(26 − d) × (26 + d) = 16302 / 26
262 − d² = 627
676 − d² = 627
676 − 627 = d2 
49 = d²

d = 7

5. What is the value of x for which the numbers 3x + 2, 7x − 3, 25 are in A.P.?

a) 4
b) 2
c) 5
d) 3

Answer: d) 3

Explanation: We know that if the numbers a, b and c are in A.P., then

b − a = c − b

2b = a + c

Given: 3x + 2, 7x − 3, 25 are in A.P.

Thus, a = 3x + 2, b = 7x − 3 and c = 25

2(7x − 3) = (3x + 2) + 25
14x − 6 = 3x + 27
14x − 3x = 27 + 6
11x = 33
x = 33 / 11
x = 3