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Mathematics

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.

AP GP

ICSE Sp 2025

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Answer

Let common difference be d.

a = 3

Sum of first n terms of an A.P. = n2(a + l)\dfrac{\text{n}}{2}(\text{a } + \text{ l})

Given,

The sum of the first 8 terms is twice the sum of the first 5 terms.

∴82(a+a8)=2×52(a+a5)⇒4[a+a+(8−1)d]=5[a+a+(5−1)d]⇒4[2a+7d]=5[2a+4d]⇒4[2×3+7d]=5[2×3+4d]⇒4[6+7d]=5[6+4d]⇒24+28d=30+20d⇒28d−20d=30−24⇒8d=6⇒d=68=34.\therefore \dfrac{8}{2}(a + a8) = 2 \times \dfrac{5}{2}(a + a5) \\[1em] \Rightarrow 4[a + a + (8 - 1)d] = 5[a + a + (5 - 1)d] \\[1em] \Rightarrow 4[2a + 7d] = 5[2a + 4d] \\[1em] \Rightarrow 4[2 \times 3 + 7d] = 5[2 \times 3 + 4d] \\[1em] \Rightarrow 4[6 + 7d] = 5[6 + 4d] \\[1em] \Rightarrow 24 + 28d = 30 + 20d \\[1em] \Rightarrow 28d - 20d = 30 - 24 \\[1em] \Rightarrow 8d = 6 \\[1em] \Rightarrow d = \dfrac{6}{8} = \dfrac{3}{4}.

Hence, common difference = 34\dfrac{3}{4}.

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