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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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