(c) Multiplying ${x^2} – ax + b = 0$by ${x^{n – 1}}$

${x^{n + 1}} – a{x^n} + b{x^{n – 1}} = 0$…..$(i)$

$\alpha ,\beta $are roots of ${x^2} – ax + b = 0$, therefore they will satisfy $(i)$ also

${\alpha ^{n + 1}} – a{\alpha ^n} + b{\alpha ^{n – 1}} = 0$…..$(ii)$

and ${\beta ^{n + 1}} – a{\beta ^n} + b{\beta ^{n – 1}} = 0$…..$(iii)$

Adding $(ii)$ and $(iii)$

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

or ${V_{n + 1}} – a{V_n} + b{V_{n – 1}} = 0$

or ${V_{n + 1}} = a{V_n} – b{V_{n – 1}} = 0$ (Given ${\alpha ^n} + {\beta ^n} = {V_n}$)

Trick : Put $n = 0$, $1,\,\,2$

${V_0} = {\alpha ^0} + {\beta ^0} = 2$, ${V_1} = \alpha + \beta = a$,

${\alpha ^2} + {\beta ^2} = {V_2} = {a^2} – 2b$

Now the option $(c)$ ==> ${V_2} = a{V_1} – b{V_0} = {a^2} – 2b$