KöMaL 2023 B5312

Let $F_k$ denote the $k$th Fibonacci number ($F_1=F_2=1$, $F_{k+1}=F_k+F_{k-1}$). Prove that $2\sum_{k=1}^nF_k^2F_{k+1}=F_nF_{n+1}F_{n+2}$ for all positive integers $n$.