How can I prove Newton's Sums?

[ForMyThunder]

Could anyone provide me with a proof for Newton's Sums?

So far, I've gotten as far as showing for a polynomial P(x)=a_nxⁿ+a_n-1x^n-1+...+a₁x+a₀ with roots x₁, x₂,..., x_n that

P(x₁)+P(x₂)+P(x₃)+...+P(x_n)=0

and so,

a_nS_n+a_n-1S_n-1+...+a₁S₁+na₀=0

and,

a_nS_n+k+a_n-1S_n+k-1+...+a₁S_k+1+a₀S_k=0

but I don't know if I'm heading in the right direction and I can't seem to go anywhere that seems to lead towards Newton's sums. Any suggestions will be much appreciated. Thanks.

 

[HallsofIvy]

What are S_n, S_n-1, Sup_{n+ k}, etc.?

 

[benorin]

ForMyThunder: http://www.mathlinks.ro/viewtopic.php?t=213867

HoI: http://www.artofproblemsolving.com/Wiki/index.php/Newton_sums

 

[ForMyThunder]

What are S_n, S_n-1, Sup_{n+ k}, etc.?

S_m = x₁^m+x₂^m+x₃^m+...x_n-1^m+x_n^m

 

[ForMyThunder]

Oh, thanks. :)