<s>[INST] Let $\alpha$ and $\beta$ be the roots of $x^2-x-1=0$. If $P_k=(\alpha)^k+(\beta)^k$, $k\ge 1$, then prove that- 
a) $P_5=11$$
b) $P_1+P_2+P_3+P_4+P_5 =26$

For the first part 
$$\alpha^5+\beta ^5$$
$$=(\alpha^3+\beta ^3)^2-2(\alpha \beta )^3$$
I found the value of $\alpha^3+\beta^3=4$
So $$16-2(-1)=18$$ which doesn’t match.
In the second part  depends on the value obtained from part 1, so I need to get that cleared up. 
I checked the computation many times, but it might end up being just that. Also, is there a more efficient way to do this?
 [/INST] if $$x^2-x-1=0~~~(1)$$ has roots as $a,b$ then $P_k=a^k+b^k,P_0=2,P_1=a+b=1$ and 
$$a^2-a-1=0~~~(2),~~ b^2-b-1=0~~~(3)$$
Multiply Eq.(2) once by $a^k$ and (3) by $b^k$. Adding these two Eqs. you get
$$(a^{k+2}+b^{k+2})-(a^{k+1}+b^{k+1})-(a^k+b^k)=0 \implies P_{k+2}= P_{k+1}+P_k~~~(4)$$
So $P_5=P_4+P_3, P_4=P_3+P_2, P_3=P_2+P_1, P_2=P_1+P_0=a+b+2=3$$ $$ \implies P_3=3+1=4, \implies P_4=4+3=7, P_5=7+4=11$
So $P_1+P_2+P_3+P_4+P_5=1+3+4+7+11=26$
</s>