Question

Prove that the sum of the series $1+3+7+15+31+....+$ upto $100$ terms is $2^{101}-102$.

Answer 1

$1+3+7+15+31+....+$ upto $100$ terms.

$=(2^1-1)+(2^2-1)+(2^3-1)+.......+(2^{100}-1)$

$=2^1+2^2+.....+2^{100}-100$

$=1+2^1+2^2+.....+2^{100}-101$

$=\frac{2^{101}-1}{2-1}-101$

$=2^{101}-102$.