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 – 2026

Step 1: Given first term of AP, a = 3

Step 2: Finding value of common diff, d:
We know that the sum of n terms of an AP, S_n = n/2 [2a + (n – 1)d]
Sum of first 8 terms:
S₈ = 8/2 [2a + (8 – 1) d]
= 4 (2a + 7 d)
= 8a + 14d
Sum of first 5 terms:
S₅= 5/2 [2a + (5 – 1) d]
= 5/2(2a + 4d)
= 5a +10d

Step 3: Let’s start by given condition: S₈ = 2 S₅
8a + 14d = 2 (5a + 10d)
8a + 14d = 10a + 20d
14d – 20 d = 10a – 8a
-6d = 2a
d = 2a/-6 = – a/3
by substituting the value of a, we get:
d = – (3)/3 = – 1
Therefore, the common difference of AP is – 1

Check: S₈ = 8a + 14d = 8(3) + 14 (-1) = 24 -14 = 10
S₅ = 5a + 10d = 5(3) + 10 (-1) = 15 -10 = 5
Since, above values satisfy the given condition, S₈ = 2 S₅, therefore our answer is correct.

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