Let and be two real numbers such that +=1 and =-1.

Question

Let $\alpha$ and $\beta$ be two real numbers such that $\alpha+\beta=1$ and $\alpha\beta=-1$. Let $p_n=\alpha^n+\beta^n$. If $p_{n-1}=11$ and $p_{n+1}=29$ for some integer $n\ge1$, then the value of $p_n^2$ is

Answer: 324

Accepted Answer

$\alpha,\beta$ are roots of $x^2-x-1=0$, so each satisfies $x^{n+1}=x^n+x^{n-1}$, giving the recurrence $p_{n+1}=p_n+p_{n-1}$.

$$29=p_n+11 \implies p_n=18 \implies p_n^2=324.$$

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