Fibonacci Sequence Fn denotes the nth term of the Fibonacci...

Fibonacci Sequence $F_{n}$ denotes the nth term of the Fibonacci sequence discussed in Section $12.1 .$ Use mathematical
induction to prove the statement.
$$
F_{1}^{2}+F_{2}^{2}+F_{3}^{2}+\cdots+F_{n}^{2}=F_{n} F_{n+1}
$$

Key Concepts

Mathematical Induction

Mathematical induction is a fundamental proof technique used to establish the truth of an infinite sequence of statements. It involves proving a base case to show that the statement holds for an initial value, and then demonstrating that if the statement holds for an arbitrary case, it necessarily holds for the next case. This method leverages the well-ordered nature of the natural numbers to extend a proven property to all relevant cases.

Fibonacci Sequence

The Fibonacci sequence is a well-known sequence of numbers where each term, after the first two, is defined as the sum of the two preceding terms. This sequence exhibits unique recursive properties and appears in various areas of mathematics, including combinatorics, number theory, and even biological settings. Its recursive definition makes it an excellent subject for proof techniques such as mathematical induction.

Base Case

The base case is the initial step in a proof by induction where the statement is shown to be true for the first instance, often the smallest number in the domain. Validating the base case is essential because it provides the necessary starting point from which the inductive process can proceed, ensuring that the subsequent steps build on a verified foundation.

Inductive Step

The inductive step involves assuming that the statement holds for a particular arbitrary case (this assumption is known as the inductive hypothesis) and then proving that the statement must also hold for the next case. This step is critical in demonstrating that if the statement is true for one case, it will continue to be true indefinitely for all subsequent cases, thereby completing the proof.

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