Question

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

Answer 1

Answer: 5

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