Question

Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $p_k=(\alpha {)}^k+(\beta {)}^k,k\ge 1$, then which one of the following statements is not true ?

• A. $(p_1+p_2+p_3+p_4+p_5)=26$
• B. $p_5=11$
• C. $p_5=p_2\cdot p_3$
• D. $p_3=p_5-p_4$

Answer 1

The correct option is C $p_5=p_2\cdot p_3$

Given $\alpha ,\beta$ are the roots of $x^2-x-1=0$
$\Rightarrow \alpha +\beta =1$ and $\alpha \beta =-1$
and ${\alpha }^2=\alpha +1$ and ${\beta }^2=\beta +1$

$p_k={\alpha }^{k-2}{\alpha }^2+{\beta }^{k-2}{\beta }^2$
$\Rightarrow p_k={\alpha }^{k-2}(\alpha +1)+{\beta }^{k-2}(\beta +1)$
$\Rightarrow p_k={\alpha }^{k-1}+{\beta }^{k-1}+{\alpha }^{k-2}+{\beta }^{k-2}$
$\Rightarrow p_k=p_{k-1}+p_{k-2}$

$p_1=1,p_2=\alpha +\beta +1+1=3$
$\Rightarrow p_3=p_2+p_1=4$
$p_4=p_3+p_2=7$
$p_5=p_4+p_3=11$

$\therefore p_5\ne p_2\cdot p_3$
and $p_1+p_2+p_3+p_4+p_5=26$
and $p_3=p_5-p_4$