## Discrete Mathematics with Applications 4th Edition

We know that, $1 + r + r^2 + ··· + r^n = \frac{r^{n+1}− 1}{r − 1}$ a) Thus, $1 + 2 + 2^2 + ··· + 2^{25}$ = $\frac{2^{25+1} − 1}{2-1}$ = $\frac{2^{26} − 1}{2-1}$ = $2^{26} − 1 = 67108863$ b. $2 + 2^2 + 2^3 + ··· + 2^{26}$ = $2\times(1 + 2 + 2^2 + ··· + 2^{25} )$ = $2\times(67108863)$ [ by (a) ] = 134217726