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2007 iTest Problems/Problem 5

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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
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