Determine whether the statement is true or false.
-For any odd natural number n: If you square the nth term of the Lucas sequence and add 2 you will obtain the (2n)th term of the Lucas sequence.

Question

Quizplus · Accepted Answer

The Lucas sequence is defined by the recurrence relation L(n) = L(n-1) + L(n-2), with initial values L(0) = 2 and L(1) = 1. The nth term of the Lucas sequence is given by L(n) = (φ^n - ψ^n) / √5, where φ = (1 + √5) / 2 and ψ = (1 - √5) / 2 are the golden ratio and its conjugate. Squaring the nth term of the Lucas sequence and adding 2 gives (L(n))^2 + 2 = (φ^n - ψ^n)^2 / 5 + 2 = (φ^(2n) - ψ^(2n)) / 5 + 2 = L(2n), which is the (2n)th term of the Lucas sequence.

Determine Whether the Statement Is True or False