First slide

Geometric progression

Question

$If S_{P} denotes the sum of the series 1 + r^{p} + r^{2 p} + \dots . to ond S_{p} denotes$ $the sum of the series 1 - r^{p} + r^{2 p} - r^{3 p} + \dots . t o \infty, | r | < 1, then S_{p} + s_{p} in terms of$ $S_{2 p} is$

Moderate

A

$2 S_{2 p}$
B

$0$
C

$\frac{1}{2} S_{2 p}$
D

$- \frac{1}{2} S_{2 p}$

Solution

$S_{P} = \frac{1}{1 - r^{p}}, s_{p} = \frac{1}{1 + r^{p}}, S_{2 p} = \frac{1}{1 - r^{2 p}}$
$Clearly, S_{P} + s_{p} = \frac{2}{1 - r^{2 p}} = 2 S_{2 p}$

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