From Lucas numbers to Fibonacci numbers

Statement

The identity $L_{n-1}+L_{n+1}=5F_n$ holds. Verify it for $n=6$.

Solution

With $n=6$: $L_5=11$ and $L_7=29$, so $L_5+L_7=11+29=\boxed{40}=5\cdot 8=5F_6.$