Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$ and $S_n={\alpha }^n+{\beta }^n$ then $a{S}_{n+1}+b{S}_n+c{S}_{n-1}=(n\ge 2)$

• A. \N
• B. $a+b+c$
• C. $(a+b+c)n$
• D. $n^{2}abc$

Answer 1

The correct option is A \N

$S_{n+1}={\alpha }^{n+1}+{\beta }^{n+1}$
$=(\alpha +\beta )({\alpha }^n+{\beta }^n)-\alpha \beta ({\alpha }^{n-1}+{\beta }^{n-1})$
$=-\frac{b}{a}.S_n-\frac{c}{a}.S_{n-1}$
$a{S}_{n+1}+b{S}_n+c{S}_{n-1}$ = $-b{S}_n-c{S}_{n-1}+b{S}_n+c{S}_{n-1}$
= 0