Product of nth Lucas and Fibonacci Numbers

Theorem

Let $L_k$ be the $k$th Lucas number.

Let $F_k$ be the $k$th Fibonacci number.

Then:

$\forall n \in \N_{>0}: F_n L_n = F_{2 n}$

Proof

By definition of Lucas numbers:

$L_n = F_{n - 1} + F_{n + 1}$

Hence:

$F_n L_n = F_n \paren {F_{n - 1} + F_{n + 1} }$

From Honsberger's Identity:

$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

The result follows by setting $m = n$.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {1-1}$ Principle of Mathematical Induction: Exercise $15$