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1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between? a)

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GK_Gmat
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1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between? a) [#permalink] New post   04 Oct 2007, 00:52
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1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between?

a) 1/2 & 2/3
b) 2/3 & 3/4
c) 3/4 & 9/10
d) 9/10 & 10/9
e) 10/9 & 3/2

Pls. explain your ans.



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Re: 1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between? a) [#permalink] New post   04 Oct 2007, 02:24
1/2+1/4 = 3/4

3/4+1/8 = 7/8

7/8+1/16 = 15/16

see the pattern here ?

so the answer is (2^20-1)/(2^20) = ~ 1

the answer is (D)

:)
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Re: 1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between? a) [#permalink] New post   04 Oct 2007, 03:25
GK_Gmat wrote:
1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between?

a) 1/2 & 2/3
b) 2/3 & 3/4
c) 3/4 & 9/10
d) 9/10 & 10/9
e) 10/9 & 3/2

Pls. explain your ans.



the sum + (1/2)^20 - (1/2)^20 = 1 - (1/2)^20.

so obviously it is D.
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Re: 1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between? a) [#permalink] New post   04 Oct 2007, 04:03
KillerSquirrel wrote:
1/2+1/4 = 3/4

3/4+1/8 = 7/8

7/8+1/16 = 15/16

see the pattern here ?

so the answer is (2^20-1)/(2^20) = ~ 1

the answer is (D)

:)


KS - I understand the pattern; I had picked C instead of D; but I guess that since the range of C is larger, D is a better estimate of the value.

However, can you pls. explain how you get (2^20-1)/(2^20)?

Thanks.
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Re: 1/2 + (1/2)^2 + (1/2)^3 + ... + (1/2)^20 is between? a) [#permalink] New post   Updated on: 04 Oct 2007, 04:20
GK_Gmat wrote:
KillerSquirrel wrote:
1/2+1/4 = 3/4

3/4+1/8 = 7/8

7/8+1/16 = 15/16

see the pattern here ?

so the answer is (2^20-1)/(2^20) = ~ 1

the answer is (D)

:)


KS - I understand the pattern; I had picked C instead of D; but I guess that since the range of C is larger, D is a better estimate of the value.

However, can you pls. explain how you get (2^20-1)/(2^20)?

Thanks.


1/2+1/4 = 3/4 = (4-1)/4 = (2^2-1)/2^2

3/4+1/8 = 7/8 = (8-1)/8 = (2^3-1)/2^3

7/8+1/16 = 15/16 = (16-1)/16 = (2^4-1)/2^4

since that last value added is 1/2^20 then (2^20-1)/2^20 the outcome is a number very close to 2^20/2^20 = 1

the pattern is:

1/2+1/2^2+1/2^3 .... 1/2^n = (2^n-1)/(2^n) for every n > 0.

:)



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Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!

Originally posted by KillerSquirrel on 04 Oct 2007, 04:12.
Last edited by KillerSquirrel on 04 Oct 2007, 04:20, edited 1 time in total.
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