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1/2 + (1/2)^2.... + (1/2)^20 is between: 1/2 and 2/3 2/3 and

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anonymousegmat
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1/2 + (1/2)^2.... + (1/2)^20 is between: 1/2 and 2/3 2/3 and [#permalink] New post   17 Jul 2007, 04:53
1/2 + (1/2)^2.... + (1/2)^20 is between:


1/2 and 2/3
2/3 and 3/4
3/4 and 9/10
9/10 and 10/9
10/9 and 3/2



this is from one of the challenges, and the explanations sucks. it says to imagine a piece of string being cut in half 20 times over... so therefore it will be less than one but close to one. um.. OK, i'd rather be able to solve to problem mathematically.

i am assuming this is a candidate for geometric progression... could someone please explain how to use it



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dahcrap
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Re: 1/2 + (1/2)^2.... + (1/2)^20 is between: 1/2 and 2/3 2/3 and [#permalink] New post   17 Jul 2007, 10:40
I simply approximated it by using the GP formula for infinite series where Sum = a/(1-r) Here a = 1/2 and r =1/2
So we get approximately 1.
So I guess the answer is D. I forgot the formula for a non infinite series though
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Himalayan
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Re: 1/2 + (1/2)^2.... + (1/2)^20 is between: 1/2 and 2/3 2/3 and [#permalink] New post   17 Jul 2007, 15:13
dahcrap wrote:
I simply approximated it by using the GP formula for infinite series where Sum = a/(1-r) Here a = 1/2 and r =1/2
So we get approximately 1.

So I guess the answer is D. I forgot the formula for a non infinite series though


is it (sum) = a (1 - r^n) / (1 - r)
= [1/2 {1 - (1/2)^20}] / [1 - 1/2]
= 1 approxxxxxxxxxxxxxx.
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Re: 1/2 + (1/2)^2.... + (1/2)^20 is between: 1/2 and 2/3 2/3 and [#permalink] New post   17 Jul 2007, 20:28
anonymousegmat wrote:
1/2 + (1/2)^2.... + (1/2)^20 is between:


1/2 and 2/3
2/3 and 3/4
3/4 and 9/10
9/10 and 10/9
10/9 and 3/2



this is from one of the challenges, and the explanations sucks. it says to imagine a piece of string being cut in half 20 times over... so therefore it will be less than one but close to one. um.. OK, i'd rather be able to solve to problem mathematically.

i am assuming this is a candidate for geometric progression... could someone please explain how to use it


(1/2)^1 = 1- (1/2)^1
(1/2)^2 = 1/2 -(1/2)^2
...
(1/2)^20 = (1/2)^19-(1/2)^20

Add all the above equations together.
1/2 + (1/2)^2.... + (1/2)^20=1-(1/2)^20
Therefore, D is the answer.



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Hi there,
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Re: 1/2 + (1/2)^2.... + (1/2)^20 is between: 1/2 and 2/3 2/3 and [#permalink]
17 Jul 2007, 20:28
Moderator:
yashikaaggarwal
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3137 posts

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