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Define $P_n$ as follows: $$P_n(a,b,c) = \frac {a^n}{(a-b)(a-c)}+\frac {b^n}{(b-a)(b-c)}+\frac {c^n}{(c-a)(c-b)}$$ with $n \in N$.

I know that $P_2(a,b,c)=1,$ and $ P_3(a,b,c)=a+b+c$.

How can I find $P_3, P_4, P_5$?

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    What are the limits of the summation? What are a, b, and c?2012-10-27
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    What is even the *index* of summation?2012-10-27
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    a,b,c is any numbers2012-10-27
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    What are you adding together? Is it three terms with one each in $a, b, c$? In which case it is easier to write out the whole thing, which also avoids possible confusion about the signs of the terms. And then when you are asking "find" what sort of answer do you want - are you aiming for elementary symmetric polynomials or something of that kind?2012-10-27
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    What is $N^*$? The only place that I've seen such notation is as the set of all (nonstandard) natural numbers in a saturated model ${\bf R}^*$ of real numbers (in nonstandard analysis).2012-10-27
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    ok there 're some difference from our country , and in our mathematic , but it is not problem I have check2012-10-28
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    I did a major edit in accordance with the discussion following my tentative answer - comments already made refer to the unedited version.2012-10-28

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