Lucas Number Questions!

Question

Problem:

Find $(a,b)$ such that $$L_n = a\phi^n + b\widehat{\phi}^n.$$

Where $n$ is the $n^{th}$ lucas number.

How would I start this? Would I just start by plugging in $a=b=1$ and then trying to solve?

Answer 1

$$L_0=2$$

$$L_1=1$$

Thus,

$$a+b=2$$

$$a\phi+b\hat\phi=1$$

And so,

$$a=2-b$$

$$(2-b)\phi+b\hat\phi=1$$

$$\implies b=\frac{1-2\phi}{\hat\phi-\phi}$$

$$\implies a=2-\frac{1-2\phi}{\hat\phi-\phi}$$