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?
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?
$$L_0=2$$
$$L_1=1$$
Thus,
$$a+b=2$$
$$a\phi+b\hat\phi=1$$
And so,
$$a=2-b$$
$$(2-b)\phi+b\hat\phi=1$$
$$\implies b=\frac{1-2\phi}{\hat\phi-\phi}$$
$$\implies a=2-\frac{1-2\phi}{\hat\phi-\phi}$$