Question

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

• A. $\left(p_{1}+p_{2}+p_{3}+p_{4}+p_{5}\right)=26$
• B. $p_5=11$
• C. $p_{5}=p_{2}\cdot p_{3}$
• D. $p_{3}=p_{5}-p_{4}$

Answer 1

Answer: (C)

$\alpha+\beta=1, \alpha\beta=-1$
$P_{k}=\alpha^{k}+\beta^{k}
\alpha^{2}-\alpha-1=0$
$\Rightarrow \alpha^{k}-\alpha^{k-1}-\alpha^{k-2}=0$
$\&\,\beta^{k}-\beta^{k-1}-\beta^{k-2}=0$
$\Rightarrow P_{k}=P_{k-1}+P_{k-2}
P_{1}-\alpha+\beta=1$
$P_{2}=\left(\alpha+\beta\right)^{2}-2\alpha\beta=1+2=3$
$P_{3}=4$
$P_{4}=7$
$P_{5}=11 $