Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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If Sp Denotes the Sum of the Series 1 + Rp + R2p + ... to ∞ and Sp the Sum of the Series 1 − Rp + R2p − ... to ∞, Prove that Sp + Sp = 2 . S2p. - Mathematics

If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.

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Solution

We have:

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

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

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

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

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

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

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

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.4 | Q 4 | Page 40
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