If *S*_{p} denotes the sum of the series 1 + r^{p} + r^{2p} + ... to ∞ and s_{p} the sum of the series 1 − r^{p} + r^{2p} − ... to ∞, prove that S_{p} + s_{p} = 2 . S_{2p}.

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#### Solution

We have:

\[ S_p = 1 + r^p + r^{2p} + . . . \infty \]

\[ \therefore S_p = \frac{1}{1 - r^p}\]

\[\text { Similarly }, s_p = 1 - r^p + r^{2p} - . . . \infty \]

\[ \therefore s_p = \frac{1}{1 - \left( - r^p \right)} = \frac{1}{1 + r^p}\]

\[\text { Now }, S_P + s_p = \frac{1}{1 - r^p} + \frac{1}{1 + r^p} = \frac{\left( 1 - r^p \right) + \left( 1 + r^p \right)}{\left( 1 - r^{2p} \right)}\]

\[ \Rightarrow \frac{2}{1 - r^{2p}} = 2 S_{2P} \]

\[ \therefore S_P + s_p = 2 S_{2P}\]

Concept: Geometric Progression (G. P.)

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