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.)

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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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.)

Compute the sum \[\sum_{k=0}^{2011}\; \frac{a_k^3}{1-3a_k+3a_k^2}\]

(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
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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No reply!!! Is it so easy or hard that nobody tries?Well I'm giving a hint:

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I think the "too easy" part is true

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Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi

- zadid xcalibured
**Posts:**217**Joined:**Thu Oct 27, 2011 11:04 am**Location:**mymensingh

pairing yeids an elegant solution.

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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I agree.*Mahi* wrote:I think the "too easy" part is true

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