Excrcise 2. Let F, be the n-th Fibonacci number. Prove E = F„Fn+1- %3D k–1
Advanced Engineering Mathematics
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Exercise 2.** Let \( F_n \) be the \( n \)-th Fibonacci number. Prove
\[
\sum_{k=1}^{n} F_k^2 = F_n F_{n+1}.
\]
This exercise involves proving a mathematical identity related to Fibonacci numbers. The equation suggests that the sum of the squares of the first \( n \) Fibonacci numbers equals the product of the \( n \)-th Fibonacci number and the \((n+1)\)-th Fibonacci number.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07bcd84f-e6c2-4c3e-8d41-6b3fbe31a270%2Fd58c9076-94e6-402c-91a7-af65c82bad8e%2Fzjus2r_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 2.** Let \( F_n \) be the \( n \)-th Fibonacci number. Prove
\[
\sum_{k=1}^{n} F_k^2 = F_n F_{n+1}.
\]
This exercise involves proving a mathematical identity related to Fibonacci numbers. The equation suggests that the sum of the squares of the first \( n \) Fibonacci numbers equals the product of the \( n \)-th Fibonacci number and the \((n+1)\)-th Fibonacci number.
Expert Solution
Step 1
Given is the Fibonacci number.
The above problem can be proved using induction hypothesis over n.
When n=1:
Thus, it is true for n=1.
When n=m
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