Question

If $S_p$ denote the sum of the series $1+r^p+r^{2p}+..$ upto infinity and $X_p$ be the sum of the series $1-r^p+r^{2p}-..$ upto infinity then $(r\in (-1,1)-\left\{0\right\})$

• A. $S_p+X_p={2X}_{2p}$
• B. $S_p+X_p={2S}_{2p}$
• C. $S_p+X_p=S_{2p}$
• D. $Noneofthese$

Answer 1

Answer: (A)

$S_p+X_p={2X}_{2p}$

$\begin{array}{c}{S}_{p}ism-1+r^p+r^{2p}...\\ X_p=1-r^p+r^{2p}\\ S_p=\frac{9}{1-R}=\frac{1}{1-r^p}\\ X_p=\frac{1}{1+r^p}\\ S_p+X_p=\frac{1}{1-r^p}+\frac{1}{1+r^p}\\ =\frac{1+r^p+1}{\left(1-r^p\right)\left(1+r^p\right)}=\frac{2}{1-r^{2p}}\\ =2S_p\end{array}$