Question

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

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

Answer 1

The correct option is D $5S_6+6S_5=2S_4$

$5x^2+6x-2=0$
$\because \alpha ,\beta$ are roots
$\therefore 5{\alpha }^2+6\alpha =2\Rightarrow 6\alpha -2=5{\alpha }^2$

Similarly

$6\beta -2=-5{\beta }^2$
$S_6={\alpha }^6+{\beta }^6$
$S_5={\alpha }^5+{\beta }^5$
$S_4={\alpha }^4+{\beta }^4$

Now

$6S_5-2S_4=6{\alpha }^5-2{\alpha }^4+6{\beta }^5-2{\beta }^4$
$=a^4(6\alpha -2)+{\beta }^4(6\beta -2)={\alpha }^4(-5{\alpha }^2)+{\beta }^4(-5{\beta }^2)=-5({\alpha }^6+{\beta }^6)=-5S_6\Rightarrow 6S_5+5S_6=2S_4$