Originally Posted by

**tonio** It'd be a great idea if you didn't assume everybody understands your notation and define what os $\displaystyle F_n$...Fibonacci sequence or what!?

And use parentheses: you actually meant $\displaystyle F_n=5F_{n-4}+3F_{n-5}$ , so if your Fibonacci sequence is

$\displaystyle 1,1,2,3,5,8,13,21,34, 55,...$, then $\displaystyle F_6=8=5F_2+3F_1=5\cdot 1+3\cdot 1$ and you have the base case.

Suppose truthness for $\displaystyle k<n$ , and take $\displaystyle F_{n+1}:=F_n+F_{n-1}$ . The inductive hypotheses tells us that both terms in the RHS fulfill the equality, so:

$\displaystyle F_{n+1}=F_n+F_{n-1}=5F_{n-4}+3F_{n-5}+5F_{n-5}+3F_{n-6}$ ...and now just rearrange these terms and use the definition of Fibonacci sequence...