**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.

Answered: Excrcise 2. Let F, be the n-th…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.4: Fractional Expressions

Problem 65E

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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.

Expert Solution

Step 1

Given $F_{n}$ is the $n^{th}$ Fibonacci number.

The above problem can be proved using induction hypothesis over n.

When n=1:

$\sum_{k = 1}^{n} {F_{k}}^{2} = {F_{1}}^{2} = {(1)}^{2} = 1 = 1 \cdot 1 = F_{1} \cdot F_{2}$

Thus, it is true for n=1.

When n=m

$\sum_{k = 1}^{m} {(F_{m})}^{2} = F_{m \cdot} F_{m + 1} . . . . . (1)$

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Excrcise 2. Let F, be the n-th Fibonacci number. Prove E = F„Fn+1- %3D k–1