Mathematics - Quadratic Equation Question with Solution | TestHub

The roots $\alpha ,\beta$ satisfy $a{\alpha }^2+b\alpha +c=0$ and $a{\beta }^2+b\beta +c=0$.

Multiply the first by ${\alpha }^{n-1}$ and the second by ${\beta }^{n-1}$:

$a{\alpha }^{n+1}+b{\alpha }^n+c{\alpha }^{n-1}=0$

$a{\beta }^{n+1}+b{\beta }^n+c{\beta }^{n-1}=0$

Adding these two equations:

$a({\alpha }^{n+1}+{\beta }^{n+1})+b({\alpha }^n+{\beta }^n)+c({\alpha }^{n-1}+{\beta }^{n-1})=0$

Using $S_k={\alpha }^k+{\beta }^k$:

$a{S}_{n+1}+b{S}_n+c{S}_{n-1}=0$

The final answer is $\boxed{0}$.