First slide

Arithmetico-Geometric Progression

Question

The sum to infinity of the series $1 + \frac{4}{5} + \frac{7}{5^{2}} + \frac{10}{5^{3}} + \dots$ is

Moderate

A

$\frac{16}{35}$
B

$\frac{11}{8}$
C

$\frac{35}{16}$
D

$\frac{17}{6}$

Solution

$S = 1 + \frac{4}{5} + \frac{7}{5^{2}} + \frac{10}{5^{3}} + \dots$ (1)
$\frac{1}{5} S = \frac{1}{5} + \frac{4}{5^{2}} + \frac{7}{5^{3}} + \dots$ (2)
Subtracting (2) from (1), we get
$\frac{4}{5} S = 1 + \frac{3}{5} + \frac{3}{5^{2}} + \frac{3}{5^{3}} + \dots = 1 + \frac{3 / 5}{1 - 1 / 5}$
(1) $= 1 + \frac{3}{4} = \frac{7}{4}$
(2)
$\Rightarrow S = \frac{35}{16}$

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