Question

If $S_p$ denote the sum of the series $1+r^p+r^{2p}+...\infty$ and

$S_p^{\prime }$ denote the sum of the series $1-r^p+r^{2p}+...\infty$${S_p}^{\prime }$, then $S_p+{S_p}^{\prime }$ equals

• A. $2S_{2p}$
• B. $2.{S_{2p}}^{\prime }$
• C. $S_p.{S_p}^{\prime }$
• D. $2S_p-{S_p}^{\prime }$

Answer 1

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

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

${S^{\prime }}_p=1-r^p+r^{2p}+...\infty =\frac{1}{1+r^p}$

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

Now $S_p+{S^{\prime }}_p=\frac{1}{1-r^p}+\frac{1}{1+r^p}=\frac{1+r^p+1-r^p}{(1-r^p)(1+r^p)}=\frac{2}{1-r^{2p}}=2S_{2p}$