Question

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

• A. $5 S _{6}+6 S _{5}=2 S _{4}$
• B. $5 S _{6}+6 S _{5}+2 S _{4}=0$
• C. $6 S _{6}+5 S _{5}+2 S _{4}=0$
• D. $6 S _{6}+5 S _{5}=2 S _{4}$

Answer 1

Answer: (A)

$\alpha$ and $\beta$ are roots of $5 x^{2}+6 x-2=0$
$\Rightarrow 5 \alpha^{2}+6 \alpha-2=0$
$\Rightarrow 5 \alpha^{n+2}+6 \alpha^{n+1}-2 \alpha^{n}=0 \ldots(1)$
(By multiplying $\left.\alpha^{n}\right)$
Similarly $5 \beta^{n+2}+6 \beta^{n+1}-2 \beta^{n}=0 \ldots$(2)
By adding (1)$\&(2)$
$5 S_{n+2}+6 S_{n+1}-2 S_{n}=0$
For $n=4$
$5 S _{6}+6 S _{5}=2 S _{4}$