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 Lm+n = fm+1Ln + fmLn−1 whenever n and n are positive integers with n > 1, fn is the nth Fibonacci number, and Ln is the nth Lucas number.
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