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Suppose $x_0$ , $x_1$ , $x_2$ , ... , $x_n$ are distinct real numbers , prove that :

$$\large{\displaystyle{\sum_{i=0}^{n}\left(x_{i}^{n}\prod_{0\leq k\leq n}^{k\neq i}\frac{x-x_k}{x_i-x_k}\right)=x^n}}$$

I have no ideas to do this question

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yes, i am sorry !That is typing error! The question correct now –  cwk709394 Oct 24 '12 at 11:58
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Both sides are polynomials of degree $\le n$ in $x$. Hence they are equal, if they are equal at $n+1$ disticnt points. Check equality at $x = x_j$, $0 \le j \le n$. –  martini Oct 24 '12 at 11:58
    
thanks for your suggestion –  cwk709394 Oct 25 '12 at 0:58

1 Answer 1

up vote 3 down vote accepted

That's simply Lagrange's polynomial interpolation formula for the values of the polynomial $x^n$. Since there are $n+1$ data points, the two polynomials coincide.

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