Explanation

(c)
Multiplying $x^{2} - ax + b = 0$ by $x^{n - 1}$
$x^{n + 1} - ax^{n} + bx^{n - 1} = 0$ ...(i)
$\alpha,\ \beta$ are the roots of $x^{2} - ax + b = 0$, therefore they will satisfy Eq. (i)
Also, $\alpha^{n + 1} - a\alpha^{n} + b\alpha^{n - 1} = 0$ …(ii)
and $\beta^{n + 1} - \alpha\beta^{n} + b\beta^{n - 1} = 0$
Adding Eqs. (ii) and (iii), we get
$\left( \alpha^{n + 1} + \beta^{n + 1} \right) - a\left( \alpha^{n} + \beta^{n} \right) + b\left( \alpha^{n - 1} + \beta^{n - 1} \right) = 0$
or $V_{n + 1} - aV_{n} + bV_{n - 1} = 0$
or $V_{n + 1} = aV_{n} - bV_{n - 1} = 0$ (Given $\alpha^{n} + \beta^{n} = V_{n})$