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…

Let common difference be d.

First term (a) = 3

By formula,

Sum of first n terms of an A.P. = $\frac{n}{2}[2a + (n - 1)d]$

Given,

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

⇒ S₈ = 2 × S₅

$\therefore \frac{8}{2}[2 \times a + (8 - 1) \times d] = \frac{5}{2}[2 \times a + (5 - 1) \times d]$

⇒ 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 = $\frac{6}{8} = \frac{3}{4}$.

Hence, common difference = $\frac{3}{4}$.