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 ___________.

We have, $S=109+\frac{108}{5}+\frac{107}{5^2}+\ldots+\frac{2}{5^{107}}+\frac{1}{5^{108}}$ ...........(i) <br/><br/>$\frac{S}{5}=\frac{109}{5}+\frac{108}{5^2}+\frac{107}{5^3}+\ldots .+\frac{2}{5^{108}}+\frac{1}{5^{109}}$ .............(ii) <br/><br/>On subtracting Eq. (ii) from Eq. (i), we get <br/><br/>$$ \begin{aligned} \frac{4 S}{5} & =109-\frac{1}{5}-\frac{1}{5^2}-\ldots .-\frac{1}{5^{108}}-\frac{1}{5^{109}} \\\\ \frac{4 S}{5} & =109-\left[\frac{1}{5}+\frac{1}{5^2}+\ldots+\frac{1}{5^{108}}+\frac{1}{5^{109}}\right] \end{aligned} $$ <br/><br/>This form a GP with $r=\frac{1}{5}$ <br/><br/>$$ \begin{aligned} \frac{4 S}{5} & =109-\frac{1}{5}\left[\frac{1-\frac{1}{5^{109}}}{1-\frac{1}{5}}\right]\\\\ &=109-\frac{1}{4}\left[1-\frac{1}{5^{109}}\right] \\\\ & =109-\frac{1}{4}+\frac{1}{4} \times \frac{1}{5^{109}} \end{aligned} $$ <br/><br/>$\therefore$ $4 S=\frac{5}{4}\left[435+\frac{1}{5^{109}}\right]$ <br/><br/>$\Rightarrow 16 S=2175+\frac{1}{5^{108}}$ <br/><br/>$16 S-(25)^{-54}=2175$