(c) Show that the sum of the first n odd-indexed Fibonacci numbers, n ΣF2k-1 = F₁+F3 + F5 + k=1 + F2n-1, always equals the n-th even-indexed Fibonacci number, F2n. d) Show that the sum of the squares of the first n Fibonacci numbers, F² + F2 + + F2, always equals Fn. Fn+1.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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=
The Fibonacci numbers are a sequence numbers defined by F₁ = 1, F₂ =
Fn = Fn-1 + Fn-2 for n ≥ 3.
1, and
Transcribed Image Text:= The Fibonacci numbers are a sequence numbers defined by F₁ = 1, F₂ = Fn = Fn-1 + Fn-2 for n ≥ 3. 1, and
(c) Show that the sum of the first n odd-indexed Fibonacci numbers,
n
ΣF2k-1 = F₁ + F3 + F5 + ·
k=1
+ F2n-1,
always equals the n-th even-indexed Fibonacci number, F2n.
(d) Show that the sum of the squares of the first n Fibonacci numbers, F² + F2 +
... + F2, always equals Fn. Fn+1.
Transcribed Image Text:(c) Show that the sum of the first n odd-indexed Fibonacci numbers, n ΣF2k-1 = F₁ + F3 + F5 + · k=1 + F2n-1, always equals the n-th even-indexed Fibonacci number, F2n. (d) Show that the sum of the squares of the first n Fibonacci numbers, F² + F2 + ... + F2, always equals Fn. Fn+1.
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