Let and be the roots of x^2-x-1=0.

Question

Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$. If $p_k=\alpha^k+\beta^k$, $k\ge1$, then which one of the following statements is not true?

Accepted Answer

Since $x^2=x+1$: $\alpha^5=5\alpha+3$, $\beta^5=5\beta+3$, giving $p_5=5(\alpha+\beta)+6=5(1)+6=11$.

Also $p_2=(\alpha+\beta)^2-2\alpha\beta=1+2=3$, $p_3=\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=1+3=4$.

$p_2\cdot p_3=12
e p_5=11$, so statement D is not true.

Solutions · 1