Let [S=109+(108)/(5)+1075^2+ .+25^107+15^108].

Question

Let $$S=109+\frac{108}{5}+\frac{107}{5^{2}}+\ldots .+\frac{2}{5^{107}}+\frac{1}{5^{108}}$$. Then the value of $$\left(16 S-(25)^{-54}\right)$$ is equal to \_\_\_\_\_\_\_\_\_\_\_.

Answer: 2175

Accepted Answer

We have, $S=109+\frac{108}{5}+\frac{107}{{5}^{2}}+\ldots +\frac{2}{{5}^{107}}+\frac{1}{{5}^{108}}$ ...........(i)

$\frac{S}{5}=\frac{109}{5}+\frac{108}{{5}^{2}}+\frac{107}{{5}^{3}}+\ldots .+\frac{2}{{5}^{108}}+\frac{1}{{5}^{109}}$ .............(ii)

On subtracting Eq. (ii) from Eq. (i), we get

$\begin{gathered} \frac{4S}{5} =109-\frac{1}{5}-\frac{1}{{5}^{2}}-\ldots .-\frac{1}{{5}^{108}}-\frac{1}{{5}^{109}} \ \ \frac{4S}{5} =109-[\frac{1}{5}+\frac{1}{{5}^{2}}+\ldots +\frac{1}{{5}^{108}}+\frac{1}{{5}^{109}}] $

This form a GP with $r=\frac{1}{5}$

$ \frac{4S}{5} =109-\frac{1}{5}[\frac{1-\frac{1}{{5}^{109}}}{1-\frac{1}{5}}] \ \ =109-\frac{1}{4}[1-\frac{1}{{5}^{109}}] \ \ =109-\frac{1}{4}+\frac{1}{4}	imes \frac{1}{{5}^{109}} \end{gathered}$

$	herefore $ $4S=\frac{5}{4}[435+\frac{1}{{5}^{109}}]$

$\Rightarrow 16S=2175+\frac{1}{{5}^{108}}$

$16S-(25{)}^{-54}=2175$

Solutions · 1