If $(w + 1)(w - 1) = w$, find $ { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } } $. Recently I was asked a question by my student that completely stumped me.

$$\text{If }(w + 1)(w - 1) = w\text{, find } { w }^{ 10 }+\frac { 1 }{ { w }^{ 10 } }. $$

One "cheat" method that I used was to solve for the exact value of $w$ from the given first equation, and then substitute it into the requested expression that we were asked to find. I got $123$ as the answer.
However, I'm quite sure there's an algebraic way to solve this. 
Anyone wants to give this a ahot?

We have $w-\frac{1}{w}=1$, and then $w^2+\frac{1}{w^2}=3$. Put $u_n=w^{2n}+\frac{1}{w^{2n}}$; we have $u_0=2$, $u_1=3$, and 
$$u_{n+1}(w^2+\frac{1}{w^2})=u_{n+2}+u_n$$
Hence $u_{n+2}=3u_{n+1}-u_n$, it is easy to compute $u_2,u_3,u_4$, and finally $u_5$.  
