Question

Let $\alpha$ and $\beta$ be the roots of the equation $5x^2+6x-2=0.$ If $S_n={\alpha }^n+{\beta }^n,n=1,2,3\dots$ then :

• A. $5S_6+6S_5=2S_4$
• B. $5S_6+6S_5+2S_4=0$
• C. $6S_6+5S_5+2S_4=0$
• D. $6S_6+5S_5=2S_4$

Answer 1

Answer: (A)

$\alpha$ and $\beta$ are roots of $5x^2+6x-2=0$
$\Rightarrow 5{\alpha }^2+6\alpha -2=0$
$\Rightarrow 5{\alpha }^{n+2}+6{\alpha }^{n+1}-2{\alpha }^n=0\dots (1)$
(By multiplying ${\alpha }^n)$
Similarly $5{\beta }^{n+2}+6{\beta }^{n+1}-2{\beta }^n=0\dots$(2)
By adding (1)$\&(2)$
$5S_{n+2}+6S_{n+1}-2S_n=0$
For $n=4$
$5S_6+6S_5=2S_4$