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 :
Correct Option: 1
Solution:
$\alpha$ and $\beta$ are roots of $5 x^{2}+6 x-2=0$
$\Rightarrow 5 \alpha^{2}+6 \alpha-2=0$
$\Rightarrow 5 \alpha^{n+2}+6 \alpha^{n+1}-2 \alpha^{n}=0$.......(1)
(By multiplying $\alpha^{\mathrm{n}}$ )
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 $n=4$
$5 \mathrm{~S}_{6}+6 \mathrm{~S}_{5}=2 \mathrm{~S}_{4}$
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