Question

Evaluate $S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+......$ to infinite terms. Find $16S$

Answer 1

$S=1+\frac{4}{5}+\frac{7}{5^2}+\frac{10}{5^3}+......$

Now, Dividing this equation by $5$

$\frac{S}{5}=\frac{1}{5}+\frac{4}{5^2}+\frac{7}{5^3}+..............$

Subtracting the Above Equations , we get

$S-\frac{S}{5}=1+\frac{3}{5}+\frac{3}{5^2}+\frac{3}{5^3}...........\Rightarrow 4\frac{S}{5}=1+3(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}.............)\Rightarrow \frac{4S}{5}=1+3\left(\frac{1/5}{1-1/5}\right)\Rightarrow \frac{4S}{5}=1+\frac{3}{4}\Rightarrow S=\frac{35}{16}\Rightarrow 16S=35$