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.
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Let common difference be d.
a = 3
Sum of first n terms of an A.P. = $\frac{\text{n}}{2}(\text{a } + \text{ l})$
Given,
The sum of the first 8 terms is twice the sum of the first 5 terms.
$$\begin{aligned}\therefore \frac{8}{2}(a + a_8) = 2 \times \frac{5}{2}(a + a_5) \\[1em] \Rightarrow 4[a + a + (8 - 1)d] = 5[a + a + (5 - 1)d] \\[1em] \Rightarrow 4[2a + 7d] = 5[2a + 4d] \\[1em] \Rightarrow 4[2 \times 3 + 7d] = 5[2 \times 3 + 4d] \\[1em] \Rightarrow 4[6 + 7d] = 5[6 + 4d] \\[1em] \Rightarrow 24 + 28d = 30 + 20d \\[1em] \Rightarrow 28d - 20d = 30 - 24 \\[1em] \Rightarrow 8d = 6 \\[1em] \Rightarrow d = \frac{6}{8} = \frac{3}{4}.\end{aligned}$$
Hence, common difference = $\frac{3}{4}$.