If $a^{3}+b^{3}+c^{3}=(a+b+c)^{3},$ prove that $a^{5}+b^{5}+c^{5}=(a+b+c)^{5}.$ Generalize your result.

Question

If $a^{3}+b^{3}+c^{3}=(a+b+c)^{3},$ prove that $a^{5}+b^{5}+c^{5}=(a+b+c)^{5} .$ Generalize your result.

Here is my attempt :–

Let $a, b, c$ be the roots of the equation $x^{3}+p x^{2}+q x+r$

$\left.\begin{array}{I}{a^{3}+pa^{2}+qa+r=0} \\ {b^{3}+pb^{2}+qb+r=0} \\ {c^{3}+pc^{2}+qc+r=0}\end{array}\right\}$

Adding the three equations,

$\sum a^{3}+p\sum a^{2}+q\sum a+3r=0$

Substituting the term given,

$(\Sigma a)^{3}+p\Sigma a^{2}+q\Sigma a+3r=0$

I can't think of anything more and I'm clueless on what to do next.

Any help would be appreciated.