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Sum of n Terms

If Sn denot...

Question

If $S_{n}$ denotes the sum of first $n$ of an $A . P$ . whose first term is $a$ and $S_{n x} / S_{x}$ is independent of $x$ , then $S_{p}$ = ?

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Solution

$∵ S_{n} = \frac{n}{2} [2 a + (n - 1) d] w h e r e a = f i r s t t e r m a n d d = c o m m o n d i f f e r e n c e h e r e, \frac{S_{n x}}{S_{x}} = \frac{\frac{n x}{2} (2 a + (n x - 1) d)}{\frac{x}{2} (2 a + (x - 1) d)} = \frac{(2 a - d) n + n^{2} x d}{(2 a - d) + x d} f o r \frac{S_{n x}}{S_{x}} b e i n d e p e n d e n t t h e n, (2 a - d) = 0 \Rightarrow d = 2 a ⟶ (1) n o w, S_{p} = \frac{p}{2} [2 a + (p - 1) d] = \frac{p}{2} [2 a + (p - 1) 2 a] (f r o m (1)) = a p^{2}$

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