2007 iTest Problems/Problem 5

Problem

Compute the sum of all twenty-one terms of the geometric series $1 + 2 + 4 + 8 + \ldots + 1048576$ .

$\mathrm{(A)}\,2097149\quad\mathrm{(B)}\,2097151\quad\mathrm{(C)}\,2097153\quad\mathrm{(D)}\,2097157\quad\mathrm{(E)}\,2097161$

Those are powers of 2. So it would make sense to multiply the series by 2:

$x=1 + 2 + 4 + 8 + \ldots + 1048576$

$2x=2 + 4 + 8 + \ldots + 1048576 + 2097152$

Subtracting,

$x=2097152-1=2097151 \Rightarrow \mathrm {(B)}$