The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers:

Question

The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: L(n) = 1 if n = 1 3 if n = 2 L(n 1) + L(n 2) if n > 2. Let = 1 + 5 2 and = 1 5 2 , as in Theorem 3.6. Prove that L(n) = n + n for all n is in N. Use strong induction. Proof. First, note that L(1) = 1 = + , and 2 + 2 = ( + 1) + + = + + 2 = = L(2). Suppose as inductive hypothesis that L(i) = i + i for all i < k, for some k > 2. Then L(k) = L(k 1) + L k = k 1 + k 1 + = k 2( + 1) + k 2 + = k 2(2) + k 2 = k + , as required.

Explanation / Answer

The question is not clear. For alpha and beta is it 1 + 52 and 1 - 52 or something else. Many terms seems missing in the question. Please take a pic of the question and post it. I'll answer it then.

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