(d) Let v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...) be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the previous two. Express v as a linear combination of the vectors (@")n>o and (B")n>0, and use this to obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.
(d) Let v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...) be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the previous two. Express v as a linear combination of the vectors (@")n>o and (B")n>0, and use this to obtain a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.
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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Question
*I only need part d)
Transcribed Image Text:(c) Let o :=
1+v5
be the golden ratio, and let ß :=
1-V5
Ey. Show that the sequences (@")n20 and (ß")n20
form a basis for W.
(d) Let
v := (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...)
be the usual sequence of Fibonacci numbers, in which each integer in the sequence is the sum of the
previous two. Express v as a linear combination of the vectors (@")n20 and (B")n>0, and use this to obtain
a closed formula for the nth Fibonacci number, i.e., the n-th term in the sequence v.
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