Question

If $S_p$ denotes the sum of the series $1+r^p+r^{2p}+.......$ to $\infty$ and $s_p$ denotes the sum of the series $1-r^p+r^{2p}-r^{3p}+.......$ to $\infty ,|r|<1$ then $S_p+s_p$ is

• A. $2S_{2p}$
• B. $0$
• C. $\frac{1}{2}{S}_{2p}$
• D. $-\frac{1}{2}{S}_{2p}$

Answer 1

The correct option is A $2S_{2p}$

Given:

$S_p=1+r^p+r^{2p}+.....\infty =\frac{1}{1-r^p}$

$s_p=1-r^p+r^{2p}-r^{3p}.....\infty =\frac{1}{1+r^p}$

$\therefore S_p+s_p$

$=\frac{1}{1-r^p}+\frac{1}{1+r^p}$

$=\frac{1+r^p+1-r^p}{(1-r^p)(1+r^p)}$

$=\frac{2}{1-r^{2p}}$

$=2S_{2p}$

$(Ans\to A)$