AoPSWiki
Trying to get to the USAMO in 2010? Our AIME Problem Series can help you get there! Click here to enroll today!

2007 iTest Problems/Problem 5

From AoPSWiki

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

Solution

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)}

See Also

2007 iTest (Problems)
Preceded by:
Problem 4
Followed by:
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Add a glimpse of the Art of Problem Solving Forum to your own site!
Click here for details!
© Copyright 2008 AoPS Incorporated. All Rights Reserved. • FoundationPrivacyContact Us