The functions $s_1=x_1+x_2+\cdots +x_n$, $s_2=\sum_{i<j} x_ix_j$, $\cdots$, $s_n=x_1x_2\cdots x_n$ are elementary symmetric functions in $x_1,x_2,\cdots,x_n$ (or more precisely, elementary symmetric polynomials).
There are other natural symmetric functions in $x_1,x_2,\cdots,x_n$: $$t_1=x_1+\cdots + x_, \hskip5mm t_2=x_1^2+x_2^2+\cdots + x_n^2, \cdots ,t_n=x_1^n+x_2^n+\cdots+x_n^n.$$ The two sets of symmetric functions are related by some identities, known as Newton's identities.
What are the simple proofs of these Newton identities? How do we prove them with some intuition or motivation?