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Algebra of Derivatives

If S n deno...

Question

If $S_{n}$ denotes the sum of $n$ terms of a G.P. whose common ratio is $r$ , then $(r - 1) \frac{d S_{n}}{d r}$ is equal to

(n - 1) S_{n} + n S_{n - 1}

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(n - 1) S_{n} - n S_{n - 1}

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(n - 1) S_{n}

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None of these

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Solution

The correct option is B $(n - 1) S_{n} - n S_{n - 1}$
We have, $S_{n} = \frac{a (r^{n} - 1)}{r - 1}$
$\Rightarrow (r - 1) S_{n} = {a r}^{n} - a$
Differentiating both sides with respect to $r,$ we get
$(r - 1) \frac{d S_{n}}{d r} + S_{n} = n a r^{n - 1} - 0$
$\Rightarrow (r - 1) \frac{d S_{n}}{d r} = n a r^{n - 1} - S_{n}$
$= n [n$ th term of $g . p .] - S_{n}$
$= n (S_{n} - S_{n - 1}) - S_{n}$
$= (n - 1) S_{n} - n S_{n - 1} .$

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