Find the $A P$ in which the ratio of the sum to $n$ terms to the sum of succeeding $n$ terms is independent of $n$ .

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Solution

Let A.P. has first term a, common difference d
Sum of n terms $= S_{n} = \frac{n}{2} (2 a + (n - 1) d)$
Sum of next terms $= S_{2 n} - S_{n} = \frac{2 n}{2} [2 a (2 n - 1) d] + \frac{D}{2} [2 a + (n - 1) d]$
Given, $y = \frac{S_{2 n} - S_{n}}{S_{n}}$ is independent of n
$\Rightarrow \frac{d y}{d n} = 0$
$\Rightarrow \frac{d}{d n} (\frac{S_{2 n} - 1}{S_{n}}) = 0$
$\Rightarrow \frac{d}{d n} [\frac{2 [2 a + (2 n - 1) d]}{[2 a + (n - 1) d]} - 1] = 0$
$\Rightarrow \frac{d}{d n} [\frac{α + 2 n d}{α + n d}] = 0$
${α = 2 a - d}$
$\Rightarrow 2 d (α + n d) - (α + 2 n d) (d) = 0$
$\Rightarrow 2 α + 2 n d - α - 2 n d = 0$
$\Rightarrow α = 0$
$\Rightarrow 2 a = d$
$\Rightarrow A . P : a, 3 a, 5 a, 7 a, . . . .$ .

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