Solving the nth number of this recurrence and cleaning it up using the binomial theorem.

Question

Given this recurrence: a_n = a_n-1 – a_n-2

I was told to create a function that would solve for a_n.

I thus came up with $a_n=\frac{\alpha^{n}-\beta^{n}}{i\sqrt{3}}$

Where $\alpha=\frac{1+i\sqrt{3}}{2}$ and $\beta=\frac{1-i\sqrt{3}}{2}$

The professor then asked the following

Your expression for $a_n$ most likely contains the imaginary number i. Use the binomial theorem to obtained a nicer expression for $a_n$.

How do I go about doing that? Thank you.

Answer 1

Note that $\alpha = e^{i\pi/3}$ and $\beta=e^{-i\pi/3}$. We then have $$\alpha^n - \beta^n = e^{in\pi/3}-e^{-in\pi/3} = 2i\sin(n\pi/3)$$ This gives us $$a_n = \dfrac{\alpha^n - \beta^n}{i\sqrt3} = \dfrac2{\sqrt3}\sin\left(n\pi/3\right)$$