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integral part of x = an integer n such that n <= x <

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integral part of x = an integer n such that n <= x < [#permalink] New post 01 Feb 2004, 02:23
[x] = integral part of x = an integer n such that n <= x < (n+1)

Find [1/3] + [2/3] + [(2^2)/3] + [(2^3)/3] + ... + [(2^1000)/3].
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 [#permalink] New post 04 Feb 2004, 01:40
(2^1001-2)/3 - 500

Can you confirm?
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AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
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MBA, Anderson School of Management, UCLA, Class of 1993

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 [#permalink] New post 04 Feb 2004, 21:24
I dont have the answer, but the following is what I've got

[(2^n)/3] = (-1/2)+(1/6)*(-1)^n +(1/3)*2^n

Answer = (-1001/2)+(1/6)+(1/3)(2^1001-1)
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 [#permalink] New post 04 Feb 2004, 22:10
AkamaiBrah wrote:
(2^1001-2)/3 - 500

Can you confirm?

I got (2^1001) / 3.

Could you explain why you are subtracting 500.
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 [#permalink] New post 05 Feb 2004, 00:12
BTW, the answer should be an integer. But 2^1001/3 is not.
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 [#permalink] New post 05 Feb 2004, 02:03
Zhung Gazi wrote:
I dont have the answer, but the following is what I've got

[(2^n)/3] = (-1/2)+(1/6)*(-1)^n +(1/3)*2^n

Answer = (-1001/2)+(1/6)+(1/3)(2^1001-1)


If you look at your answer closely, you will see that it is the same as mine.
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AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

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 [#permalink] New post 05 Feb 2004, 03:47
kpadma wrote:
AkamaiBrah wrote:
(2^1001-2)/3 - 500

Can you confirm?

I got (2^1001) / 3.

Could you explain why you are subtracting 500.


Simplest way to solve this is to assume geometry series of 1001 terms.

If you examine terms, you will notice that rounding every odd and even term subtracts 1/3 and 2/3 respectively. Hence, the answer is the sum of the series or 1/3*(2^1001-1) less the sum of the rounding 500(1/3+2/3) + 1/3 which simplifies to the answer I gave.
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Best,

AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

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 [#permalink] New post 05 Feb 2004, 08:12
AkamaiBrah wrote:
Can you confirm?
I got (2^1001) / 3.

Could you explain why you are subtracting 500.

Simplest way to solve this is to assume geometry series of 1001 terms.



My Bad, I didn't read the question stem properly
I've interpreted [X] as just ().

[x] = integral part of x = an integer n such that n <= x < (n+1)
is the key !

Thanks Akamai
  [#permalink] 05 Feb 2004, 08:12
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