Question:

Let $S=\dfrac{10^9}{5}+\dfrac{10^8}{5^2}+\cdots+\dfrac{2}{5^{108}}+\dfrac{1}{5^{108}}$. Then $(16S-(25)^{-54})$ is equal to

Step-by-Step Solution

Key Concept: Multiply by $\frac{1}{5}$ and subtract: $-\frac{4S}{5}=-10^9+\frac{1}{4}(1-5^{-109})$. Solve for $S$, then compute $16S-(25)^{-54}$.

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

Correct Answer: 2175