Mathematics - Quadratic Equation Question with Solution | TestHub

Since $\alpha ,\beta$ are roots of $a{x}^2+bx+c=0$, we have $a{\alpha }^2+b\alpha +c=0$ and $a{\beta }^2+b\beta +c=0$.

Multiplying by ${\alpha }^{n-1}$ and ${\beta }^{n-1}$ respectively:

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

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

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 $S_n={\alpha }^n+{\beta }^n$:

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