First slide

Arithmetic progression

Question

$If S_{n} denotes the sum of first n terms of an A.P whose first term is a and \frac{S_{n x}}{S_{x}} is independent of x, then S_{ρ} =$

Moderate

A

$p^{3}$
B

$p^{2} a$
C

$p a^{2}$
D

$a^{3}$

Solution

$\begin{array}{l} \begin{array}{ll} \frac{S_{n x}}{S_{x}} = \frac{\frac{n x}{2} [2 a + (n x - 1) d]}{\frac{x}{2} [2 a + (x - 1) d]} \\ = \frac{n [(2 a - d) + n x d]}{(2 a - d) + x d} \end{array} \\ For \frac{S_{n x}}{S_{x}} to be independent of x, \end{array}$
$\begin{array}{l} 2 a - d = 0 \Rightarrow 2 a = d \\ Now, \\ S_{p} = \frac{p}{2} [2 a + (p - 1) d] = p^{2} a \end{array}$

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