So I am trying to prove
$$\sum_{i=0}^n{nCi×F_i} = F_{2n}$$ Such that $$nCi = \frac {n!}{i!×(n-i)!}$$ And $F_i$ is the ith value of the fibonacci sequence such that $F_0 = 1$ and $F_1 = 1$
I have stared at this problem maybe 3 hours so far and I have found so far $$\sum_{i=0}^{n/2}{(n-i)Ci} = F_{n+1}$$
and that
$$F_{2n} = F_{n+1} × F_n + F_n × F_{n-1}$$
but so far I haven't quite found a way to use these patters for a proof. Am I on the right track? If not could anyone spare a student a hint? :).
Thank you
Also I apologize for the poor format of my math expressions, I am still new to the site and I couldn't get any syntax I personallu knew to format.