## Discrete Mathematics with Applications 4th Edition

$5\times\frac{5^{k}-25}{4}$
We know that, $1 + r + r^2 + ··· + r^n = \frac{r^{n+1}− 1}{r − 1}$ $5^3 + 5^4 + 5^5 + ··· + 5^k$ = $5^3(1+5^1 + 5^2 + ··· + 5^{k-3})$ =$5^3\times\frac{5^{(k-3)+1}-1}{5-1}$ =$5^3\times\frac{5^{k-2}-1}{4}$ or $\frac{5^{k+1}-5^3}{4}$ =$5\times\frac{5^{k}-5^2}{4}$