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2. Recall that the Fibonacci sequence a1, a2, a3,... is defined by a₁ = a₂ = 1 and an = an-1 + An-2 for all n ≥ 3. In this exercise, we will use determinants to prove the Cassini identity an+1an-1-a²=(−1)″ for all n ≥ 2. (a) Define suitable values for ao and a-1 so that the relation an = an-1 + an-2 holds for all n ≥ 1. = - (₁¹) (b) Let A: Show that an+k an+k+1, for all k-1 and all n ≥ 0. (c) Use (b) to show that = An an-1 (a an :) an+1 an Then take the determinant on both sides to deduce the Cassini identity. = An ak ak+1 ao a-1 a1 ao

2. Recall that the Fibonacci sequence a1, a2, a3,... is defined by a₁ = a₂ = 1 and an = an-1 + An-2 for all n ≥ 3. In this exercise, we will use determinants to prove the Cassini identity an+1an-1-a²=(−1)″ for all n ≥ 2. (a) Define suitable values for ao and a-1 so that the relation an = an-1 + an-2 holds for all n ≥ 1. = - (₁¹) (b) Let A: Show that an+k an+k+1, for all k-1 and all n ≥ 0. (c) Use (b) to show that = An an-1 (a an :) an+1 an Then take the determinant on both sides to deduce the Cassini identity. = An ak ak+1 ao a-1 a1 ao

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Recall that the Fibonacci sequence a₁, a2, a3,... is defined by a₁ = a₂ = 1 and An = An-1 + An-2 for all n ≥ 3. In this exercise, we will use determinants to prove the Cassini identity an+1ªn-1-a² = (−1)n for all n ≥ 2. Define suitable values for ao and a_1 so that the relation an = an−1 + An−2 holds for all n ≥ 1. (b) Let A = 01 (11) Show that an+k an+k+1= for all k-1 and all n ≥ 0. (c) Use (b) to show that An An+1 An-1 :) = = Then take the determinant on both sides to deduce the Cassini identity. = An An ak Ak+1 An ao a-1 a1 ao
Transcribed Image Text:2. Recall that the Fibonacci sequence a₁, a2, a3,... is defined by a₁ = a₂ = 1 and An = An-1 + An-2 for all n ≥ 3. In this exercise, we will use determinants to prove the Cassini identity an+1ªn-1-a² = (−1)n for all n ≥ 2. Define suitable values for ao and a_1 so that the relation an = an−1 + An−2 holds for all n ≥ 1. (b) Let A = 01 (11) Show that an+k an+k+1= for all k-1 and all n ≥ 0. (c) Use (b) to show that An An+1 An-1 :) = = Then take the determinant on both sides to deduce the Cassini identity. = An An ak Ak+1 An ao a-1 a1 ao
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