First slide

Geometric progression

Question

$The sum of the infinite series \frac{5}{13} + \frac{55}{13^{2}} + \frac{555}{13^{3}} + \dots is$

Moderate

A

$\frac{31}{18}$
B

$\frac{65}{32}$
C

$\frac{65}{36}$
D

$\frac{75}{36}$

Solution

$let S = \frac{5}{13} + \frac{55}{13^{2}} + \frac{555}{13^{3}} + \dots . - - - - (1)$
$\begin{array}{l} \frac{S}{13} = \frac{5}{13^{2}} + \frac{55}{13^{3}} + . . . . . - - - - - (2) \\ (1) - (2) \Rightarrow \frac{12}{13} S = \frac{5}{13} + \frac{50}{13^{2}} + \frac{500}{13^{3}} + \dots \\ r = 10 / 13 \\ S = \frac{13}{12} \times [\frac{\frac{5}{13}}{1 - \frac{10}{13}}] (s u m o f i n f i n i t e t e r m s i n G . P = \frac{a}{1 - r}) \\ = 65 / 36 \end{array}$

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