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1+2+2^2+2^3+2^4+2^5=？

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by Max@Math Revolution » Tue Jul 26, 2016 5:33 pm
1+2+2^2+2^3+2^4+2^5=？
A. (2^3-1)(2^3+1)
B. 2^6+1
C. 2^5-1
D. 2^5+1
E. 2^5-2

*An answer will be posted in 2 days
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by Max@Math Revolution » Thu Aug 04, 2016 3:38 am
From 1+2+2^2+2^3+2^4+2^5=1(2^6-1)/(2-1)=2^6-1=(2^3-1)(2^3+1), the correct answer is A.
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by danielle07 » Thu Aug 31, 2017 3:35 am
The answer would be

A. (2^3-1)(2^3+1)
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by Matt@VeritasPrep » Thu Aug 31, 2017 4:22 pm
Here's a little hocus pocus:

63 =>

2� - 1 =>

(2� + 2¹ + 2² + 2³ + 2� + 2�) * (2 - 1) =>

2� + 2¹ + 2² + 2³ + 2� + 2�
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by Matt@VeritasPrep » Thu Aug 31, 2017 4:24 pm
The way to solve this under test conditions, however, is to look for a pattern.

1 + 2 = 3 = 2*2 - 1

1 + 2 + 4 = 7 = 2*2*2 - 1

1 + 2 + 4 + 8 = 15 = 2*2*2*2 - 1

Aha! If I add the first n nonnegative powers of 2 on the left, I come out with 2� - 1 on the right. Under test conditions, I'll just assume that this is true and roll with it, without bothering to discover why (and on the GMAT, this will work out for me most of the time).
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