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The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: {-1) 3 Let a = L(n) = 1 + √5 2 L(n-1)+L(n-2) 1- √5 2 and B = if n = 1 if n = 2 if n > 2. as in Theorem 3.6. Prove that L(n) = a + " for all n E N. Use strong indu Proof. First, note that

The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: {-1) 3 Let a = L(n) = 1 + √5 2 L(n-1)+L(n-2) 1- √5 2 and B = if n = 1 if n = 2 if n > 2. as in Theorem 3.6. Prove that L(n) = a + " for all n E N. Use strong indu Proof. First, note that

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: if n = 1 if n = 2 L(n-1)+L(n-2) if n > 2. Let a = L(n) = 1+ √5 2 and 1 3 and B = Proof. First, note that L(1) = 1 = a + ß, a² + B² = (a + 1) + as required. 1- √5 2 = = a + ß + 2 =L(2). Suppose as inductive hypothesis that L(i) = a¹ + ßi for all i<k, for some k > 2. Then L(K) = L(k-1) + k- II = as in Theorem 3.6. Prove that L(n) = a +ß" for all n E N. Use strong induction. B + = k-1 + ßk-1 + = ak -2(a + 1) + Bk - 2 (B+ ak + = ak-2(a²) + ßk-
Transcribed Image Text:The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: if n = 1 if n = 2 L(n-1)+L(n-2) if n > 2. Let a = L(n) = 1+ √5 2 and 1 3 and B = Proof. First, note that L(1) = 1 = a + ß, a² + B² = (a + 1) + as required. 1- √5 2 = = a + ß + 2 =L(2). Suppose as inductive hypothesis that L(i) = a¹ + ßi for all i<k, for some k > 2. Then L(K) = L(k-1) + k- II = as in Theorem 3.6. Prove that L(n) = a +ß" for all n E N. Use strong induction. B + = k-1 + ßk-1 + = ak -2(a + 1) + Bk - 2 (B+ ak + = ak-2(a²) + ßk-
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