Simple but forgot the logic

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Simple but forgot the logic

by moneyman » Sun May 04, 2008 2:57 am
2+2+2^2+2^3+2^4+2^5+2^6+2^7+2^8=


2^9
2^10
2^11
2^12
2^13

Ans A
Maxx

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by VP_RedSoxFan » Sun May 04, 2008 8:20 am
If you don't know where to start, start where you know and pull out a common factor to get going.

(1) Ans = 2(1 + 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7)
(2) =2(4 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7)
(3) =2(2^2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7)

You can see that the 2 2^2 terms equal 8 which is 2^3, so:

(4) =2(2^3 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7)

Factor again with 2^3:

(5) =2^4 (1 + 1 + 2 + 2^2 + 2^3 + 2^4)

Apply same logic as before in eqn (4) to combine the 1 + 1 + 2 + 2^2 = 2^3:

(6) =2^4 (2^3 + 2^3 + 2^4); factor
(7) =2^7 (1 + 1 + 2)
(8) =2^7 (2^2)
(9) =2^9

Hope this helps, good luck
Ryan S.
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by VP_RedSoxFan » Sun May 04, 2008 8:40 am
I hit submit before I meant to finish. The above response is what I'd do rather than freezing up and cursing the GMAT if I couldn't think of anything else to do. Alternatively, you could notice that:

{1} 2^x + 2^x = 2 (2^x) = 2^(x+1)

Here you have

2 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8
=2^1 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8

using the above remark {1} we can make it

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

and repeat while collapsing:

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

until you get:

=2^8 + 2^8 = 2^9

Hope this helps as well.
Ryan S.
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by moneyman » Sun May 04, 2008 9:00 am
Thanks Ryan!! The second explanation gave me a clear shot..
Maxx