The Lucas numbers , named after Franfois-Eduoard-Anatole Lucas are defined recursively by L n =L n-1 + L n-2 , n≥3 with L 1 = 1 and L 2 = 3 . They satisfy the same recurrence relation as the Fibonacci...

TheLucas numbers, named after Franfois-Eduoard-Anatole Lucas
are defined recursively by

L_n
=L_n-1
+ L_n-2, n≥3

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

L_2n
− L_n+1L_n-1
= 5(−1)ⁿ
, n≥2.