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2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^

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alimad
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2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   29 Jun 2008, 14:56
2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 =

2 ^ 9
2 ^ 12
2 ^ 15
2 ^ 35
2 ^ 37


Please help me understand your approach.



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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   29 Jun 2008, 15:05
Write the sum of the first two terms as a power of 2
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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   29 Jun 2008, 15:09
2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^ 8

2 ^2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^ 8

kevin, I'm still lost , had a long day. Please help.
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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   29 Jun 2008, 15:20
Expert Reply
"Write the sum of the first two terms as a power of 2". Do it again with your new sum: 2^2 + 2^2 = 2*2^2 = 2^3. Then do it again... you should see the pattern.
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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   29 Jun 2008, 15:25
ahhaa, I see it now. you guys rock. Thanks
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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   30 Jun 2008, 05:31
Alternative approach (if you remember the formula for this type of series):

2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 1+ (1+ 2^1+ 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8) = [use the formula for the sum of geometric series] = 1+ (2^9-1)/1 = 2^9.

Yet another approach (from common sense when looking at the range of answers):

Each term in the sum is less than the last one which is 2^8, and overall there are 9 terms. Thus, sum < 9*2^8 < 16*2^8 = 2^4*2^8=2^12. Even such a crude estimate works for given answer choices. Only A satisfies this condition.
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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink] New post   01 Jul 2008, 06:22
I found that it is easy to just double your way to the answer:

2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 =

2+2+4+8+16+32+64+128+256 = You see from this string of numbers (and if you know binary at all) that all previous digits add up to 256. 256 + 256 = 2^9. As stated above you don't even need to add them up completely to see that it won't approach 2 ^ 12.



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Re: 2 + 2 + 2 ^2 + 2 ^3 + 2 ^4 + 2 ^5 + 2 ^6 + 2 ^7 + 2 ^8 = 2 ^ [#permalink]
01 Jul 2008, 06:22
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