Question

Evaluate $\sum _{n=1}^{50}(2^n-1)$.

Answer 1

$\sum _{n=1}^{50}(2^n-1)$

$=\sum _{n=1}^{50}{2}^n-\sum _{n=1}^{50}\left(1\right)$

Now $\sum _{n=1}^{50}{2}^n=2+2^2+2^3+2^4+2^5$

$\therefore \sum _{n=1}^{50}{2}^n=\frac{2(2^{50}-1)}{2-1}=2^{51}-2$

Now $\sum _{n=1}^{50}\left(1\right)=1+2+3+4+5$

$=\frac{50}{2}(2+(50-1\left)\right)$

$=25[2+49]=25\times 51$

$\therefore \sum _{n=1}^{50}\left(1\right)=1275$

$\therefore \sum _{n=1}^{50}(2^n-1)=2^{51}-2-1275=2^{51}-1277$

Answer is $2^{51}-1277$