Expressing in terms of symmetric polynomials. How to express $$a^7+b^7+c^7$$  in terms of symmetric polynomials ${\sigma}_{1}=a+b+c$, ${\sigma}_{2}=ab+bc+ca$ and ${\sigma}_{3}=abc$ ?

Hint:
$$a^n+b^n+c^n=(a+b+c)(a^{n-1}+b^{n-1}+c^{n-1})-(ab+bc+ca)(a^{n-2}+b^{n-2}+c^{n-2})+abc(a^{n-3}+b^{n-3}+c^{n-3}), \forall n\in\mathbb N_{\ge 3}$$
$$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)$$
Using this you can express $a^n+b^n+c^n, \forall n\in\mathbb N$ in terms of symmetric polynomials ${\sigma}_{1}=a+b+c$, ${\sigma}_{2}=ab+bc+ca$ and ${\sigma}_{3}=abc$.
