The sum to infinity of the series  1+45+752+1053+… is

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

1. A

$\frac{16}{35}$

2. B

$\frac{11}{8}$

3. C

$\frac{35}{16}$

4. D

$\frac{17}{6}$

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### 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)

(2)

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