I wanna prove following equation
$ \sum\_{i=1}^n \prod\_{k=1,k\neq i}^n \prod\_{j=1,j\neq k}^{n+1}(x\_j - x\_k) = -\prod\_{i=1}^n \prod\_{j=1,j\neq i}^n (x\_j - x\_i) $


I have verified several numbers such as $n=2,3,4$, and try to prove it using mathematical induction, however I can't extend the 2 $\sum \sum$. Some guru told me to try resultant, but it seems to be sums of resultants, and can't be simply used.


Can anyone help me on this? thanks a lot.



Help me on proof of an equation.
$\sum\_{i=1}^n \prod\_{k=1,k\neq i}^n \prod\_{j=1,j\neq k}^{n+1}(x\_j - x\_k) = -\prod\_{i=1}^n \prod\_{j=1,j\neq i}^n (x\_j - x\_i)$


Denoting $A\_i=\prod\_{j=1,j\neq i}^n (x\_j - x\_i)$ we have


$$\sum\_{i=1}^n \prod\_{k=1,k\neq i}^n (x\_{n+1}-x\_k)A\_k = -\prod\_{i=1}^n A\_i$$


From here
$$\sum\_{i=1}^n \frac{\prod\_{k=1,k\neq i}^n (x\_{n+1}-x\_k)}{A\_i} = -1$$


The last identity is just the Lagrange interpolation polynomial (<http://en.wikipedia.org/wiki/Lagrange_polynomial>).


