Question

If $S=\frac{3}{5}+\frac{10}{5^2}+\frac{21}{5^3}+\frac{36}{5^4}+\frac{55}{5^5}+\ldots \ldots \ldots \infty$ then $4 S$ is equal to

Answer 1

$S =\frac{3}{5}+\frac{10}{5^2}+\frac{21}{5^3}+\frac{36}{5^4}+\frac{55}{5^5}+\ldots \ldots \infty$
$\frac{ S }{5}=\frac{3}{5^2}+\frac{10}{5^3}+\frac{21}{5^4}+\frac{36}{5^5} \ldots . . .$
subtracting
$\frac{4}{5} S=\frac{3}{5}+\frac{7}{5^2}+\frac{11}{5^3}+\frac{15}{5^4}+\frac{19}{5^5} $
$\frac{4}{5^2} S=\frac{3}{5^2}+\frac{7}{5^3}+\frac{11}{5^4}+\ldots$
subtracting
$\frac{16}{25} S =\frac{3}{5}+\frac{4}{5^2}+\frac{4}{5^3}+\frac{4}{5^4}+\ldots $
$=\frac{3}{5}+\frac{\frac{4}{5^2}}{1-\frac{1}{5}}=\frac{3}{5}+\frac{1}{5}$
$\frac{16}{25} s =\frac{4}{5} $
$ s =\frac{5}{4} \Rightarrow 4 s =5$