Solve the following

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 :

  1. (1) $6 S_{6}+5 S_{5}=2 S_{4}$

  2. (2) $6 S_{6}+5 S_{5}+2 S_{4}=0$

  3. (3) $5 S_{6}+6 S_{5}=2 S_{4}$

  4. (4) $5 S_{6}+6 S_{5}+2 S_{4}=0$


Correct Option: , 3

Solution:

Since, $\alpha$ and $\beta$ are the roots of the equaton

$5 x^{2}+6 x-2=0$

Then, $5 \alpha^{2}+6 \alpha-2=0,5 \beta^{2}+6 \beta-2=0$

$5 \alpha^{2}+6 \alpha=2$

$5 S_{6}+6 S_{5}=5\left(\alpha^{6}+\beta^{6}\right)+6\left(\alpha^{5}+\beta^{5}\right)$

$=\left(5 \alpha^{4}+6 \alpha^{5}\right)+\left(5 \beta^{6}+6 \beta^{5}\right)$

$=\alpha^{4}\left(5 \alpha^{2}+6 \alpha\right)+\beta^{4}\left(5 \beta^{2}+6 \beta\right)$

$=2\left(\alpha^{4}+\beta^{4}\right)=2 S_{4}$

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