Product of Fibonacci and Lucas numbers

Statement

Fibonacci numbers are $F_1=F_2=1,\ F_n=F_{n-1}+F_{n-2}$; Lucas numbers are $L_1=1,\ L_2=3,\ L_n=L_{n-1}+L_{n-2}$. The identity $F_n L_n=F_{2n}$ holds. Verify it for $n=6$.

Solution

We have $F_6=8$ and $L_6=18$, so $F_6\,L_6=8\cdot 18=\boxed{144}.$ This equals $F_{12}=144$, agreeing with $F_nL_n=F_{2n}$.