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Maths-

General

Easy

Question

If $S subscript p end subscript$ denotes the sum of the series $1 minus r to the power of p end exponent plus r to the power of 2 p end exponent minus r to the power of 3 p end exponent plus midline horizontal ellipsis$ to $infinity$ and $S subscript p end subscript$ the sum of the series $1 minus r to the power of p end exponent plus r to the power of 2 p end exponent minus r to the power of 3 p end exponent plus horizontal ellipsis$ to $infinity comma blank open vertical bar r close vertical bar less than 1$ , then $S subscript p end subscript plus s subscript p end subscript$ in terms of $S subscript 2 p end subscript$ is

$2 S_{2 p}$
0
$\frac{1}{2} S_{2 p}$
$- \frac{1}{2} S_{2 p}$

The correct answer is: $2 S subscript 2 p end subscript$

$S subscript p end subscript equals fraction numerator 1 over denominator 1 minus r to the power of p end exponent end fraction comma blank s subscript p end subscript equals fraction numerator 1 over denominator 1 plus r to the power of p end exponent end fraction comma blank S subscript 2 p end subscript equals fraction numerator 1 over denominator 1 minus r to the power of 2 p end exponent end fraction blank$
Clearly, $S subscript p end subscript plus s subscript p end subscript equals fraction numerator 2 over denominator 1 minus r to the power of 2 p end exponent end fraction equals 2 S subscript 2 p end subscript$

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