Compute the Sum

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Phlembac Adib Hasan
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Compute the Sum

Unread post by Phlembac Adib Hasan » Wed Feb 01, 2012 9:16 pm

Let $a_0,a_1,a_2,...,a_{2011}$ is a series defined as \[a_n=\frac{n}{2011}\; \; for\; \; n\ge 0\]
Compute the sum \[\sum_{k=0}^{2011}\; \frac{a_k^3}{1-3a_k+3a_k^2}\]
Without using calculator or computer.


(It's one of the coolest APMO problems.Actually I needed only $5$ minutes to solve.So I think it will be a easy problem for Higher Secondary.)
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Phlembac Adib Hasan
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Re: Compute the Sum

Unread post by Phlembac Adib Hasan » Sun Feb 12, 2012 10:44 pm

No reply!!! Is it so easy or hard that nobody tries?Well I'm giving a hint:
Use gaussian pairing tool.Also notice \[a^3-3a^2b+3ab^2=(a-b)^3+b^3\]
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*Mahi*
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Re: Compute the Sum

Unread post by *Mahi* » Mon Feb 13, 2012 12:11 pm

I think the "too easy" part is true ;)
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zadid xcalibured
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Re: Compute the Sum

Unread post by zadid xcalibured » Mon Feb 13, 2012 1:03 pm

pairing yeids an elegant solution.

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Phlembac Adib Hasan
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Re: Compute the Sum

Unread post by Phlembac Adib Hasan » Mon Feb 13, 2012 2:51 pm

*Mahi* wrote:I think the "too easy" part is true ;)
I agree. :lol: :lol: :lol:
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