Let's start from here:
$F[n] := \cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^n$
If we prove this, we are done:
$F[n+20]-F[n]=10945.F[n]+6765F[n−1]$
Do this in Mathematica:
Simplify[f[n + 20] - f[n] - 10945 f[n] - 6765 f[n - 1]]
and you get:
0
Done. :-)
One side note: It really depends on how you define things. You can't get rid of induction when you define the sequence by Induction. But if you define it from the start with no induction, you can prove some of its features without induction. So, be pragmatic.