The Lucas numbers, named after François-Eduoard-Anatole Lucas (see Chapter 7 for a biograp...

The Lucas numbers, named after François-Eduoard-Anatole Lucas (see Chapter 7 for a biography), are defined recursively by

with L1 = 1 and L2 = 3. They satisfy the same recurrence relation as the Fibonacci numbers, but the two initial values are different.

Prove that 5fn+ 1 = Ln + Ln+2 whenever n is a positive integer, fn is the nth Fibonacci number, and Ln is the nth Lucas number.