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

$$S = 109 + \frac{108}{5} + \frac{107}{5^2} + \cdots + \frac{1}{5^{108}}$$

$$S = \sum_{k=0}^{108}\frac{109-k}{5^k}$$

Compute $$S - \frac{S}{5}$$

$$\frac{4S}{5} = 109 - \frac{1}{5} - \frac{1}{5^2} - \cdots - \frac{1}{5^{108}} - \frac{1}{5^{109}}$$

$$= 109 - \frac{\frac{1}{5}(1 - \frac{1}{5^{109}})}{1 - \frac{1}{5}} = 109 - \frac{1 - 5^{-109}}{4}$$

$$S = \frac{5}{4}\left(109 - \frac{1}{4} + \frac{5^{-109}}{4}\right) = \frac{5}{4} \cdot \frac{435 + 5^{-109}}{4} = \frac{5(435 + 5^{-109})}{16}$$

$$16S = 5(435 + 5^{-109}) = 2175 + 5 \cdot 5^{-109} = 2175 + 5^{-108} = 2175 + 25^{-54}$$

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