Q) An Arithmetic Progression (A.P.) has 3 as its first term. The sum of the first 8 terms is twice the sum of the first 5 terms. Find the common difference of the A.P.

ICSE Specimen Question Paper (SQP)2025

Ans: 

It is given that first term, a = 3 (given). Let’s consider the common difference as d.

Step 1: We know that sum of first n terms of an A.P.  S_n = (2 a + (n – 1) d)

∴  Sum of first 8 terms, S₈ = (2 a + (8 – 1) d) = 4 (2 a + 7 d)

Similarly, Sum of first 5 terms, S₅ = (2 a + (5 – 1) d) = 5 (a + 2 d)

Step 2: It is given that sum of the first 8 terms is twice the sum of the first 5 terms

∴  S₈ =  2 x S₅

∴ 4 (2 a + 7 d) = 2 x 5 (a + 2 d)

∴ 2 (2 a + 7 d) = 5 (a + 2 d)

∴ 4 a + 14 d = 5 a +10 d

∴ 14 d – 10 d = 5 a – 4 a

∴ a = 4 d ……. (i)

Step 3: By substituting the value of a = 3 in equation (i), we get:

∵ a = 4 d

∴ d =

∴ d =

Therefore, the common difference is .

Check: At a = 3, d = 3/4:
S₈ = 4 (2 a + 7 d) = 8 a + 28 d = 8 x 3 + 28 x 3/4 = 24 + 21 = 45
and S₅ = 5 (a + 2 d) = 5a + 10 d = 5 x 3 + 10 x 3/4 = 15 + 15/2 = 45/2
Since 45 = 2 x 45/2 or S₈ = 2 x S₅, it satisfies the given condition in the question. Hence our solution is correct.

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