Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$ and $v_n={\alpha }^n+{\beta }^n$, then

• A. $v_{n+1}={av}_n-{bv}_n$
• B. $v_{n+1}={bv}_n-{av}_{n-1}$
• C. $v_{n+1}={av}_n-{bv}_{n-1}$
• D. None of the above

Answer 1

The correct option is C $v_{n+1}={av}_n-{bv}_{n-1}$

Given equation is $x^2-ax+b=0$ and $v_n={\alpha }^n+{\beta }^n$
$\Rightarrow x^2=ax-b$
Multiplying with $x^{n-1}$ on both sides
$\Rightarrow x^{n+1}=a{x}^n-b{x}^{n-1}$
Substituting $\alpha ,\beta$ in the above equation and adding
$\Rightarrow {\alpha }^{n+1}+{\beta }^{n+1}=a({\alpha }^n+{\beta }^n)-b({\alpha }^{n-1}+{\beta }^{n-1})$
$\therefore v_{n+1}={av}_n-{bv}_{n-1}$