Question

If $\alpha ,\beta$ are roots of equation $x^2-4x-3=0$ and $s_n={\alpha }^n+{\beta }^n,n\in N$, then the value of $\frac{s_7-4s_6}{s_5}$ is

• A. $3$
• B. $4$
• C. $5$
• D. $7$

Answer 1

The correct option is A $3$

We have,
${\alpha }^2=4\alpha +3$
Multiplying by ${\alpha }^{n-2}$
${\alpha }^n=4{\alpha }^{n-1}+3{\alpha }^{n-2}$
Similarly,
${\beta }^n=4{\beta }^{n-1}+3{\beta }^{n-2}$
Let $n=7$
${\alpha }^7+{\beta }^7=4({\alpha }^6+{\beta }^6)+3({\alpha }^5+{\beta }^5)$
$\Rightarrow S_7=4S_6+3S_5$
Hence,
$S_7-4S_6=3S_5$
$\Rightarrow \frac{S_7-4S_6}{S_5}=3$
Hence, option A is correct.