Problem 31 - Algebra and Trigonometry

Question

\(F_{n}\) denotes \(n t h\) term of the Fibonacci sequence discussed in Section \(13.1 .\) Use mathematical induction to prove the statement. $$ F_{1}^{2}+F_{2}^{2}+F_{3}^{2}+\cdots+F_{n}^{2}=F_{n} F_{n+1} $$

Step-by-Step Solution

Answer

The statement is proven by induction for all \( n \).

Step 1: Base Case

We first verify the formula for the initial value of the sequence. For \( n = 1 \), the left side is \( F_1^2 = 1^2 = 1 \), and the right side is \( F_1 F_2 = 1 \times 1 = 1 \). Both sides are equal, so the base case holds true.

Step 2: Inductive Hypothesis

Assume that the statement is true for some integer \( k \), that is, \( F_{1}^{2}+F_{2}^{2}+F_{3}^{2}+\cdots+F_{k}^{2}=F_{k} F_{k+1} \). This assumption is known as the inductive hypothesis.

Step 3: Inductive Step

We now prove the statement for \( n = k + 1 \). By the inductive hypothesis, we have:\[ F_{1}^{2}+F_{2}^{2}+...+F_{k}^{2} = F_{k} F_{k+1} \] We need to show that:\[ F_{1}^{2}+F_{2}^{2}+...+F_{k}^{2}+F_{k+1}^{2}=F_{k+1} F_{k+2} \]

Step 4: Expand and Substitute

Using the inductive hypothesis, we substitute as follows:\[ F_{k} F_{k+1} + F_{k+1}^2 = F_{k+1} F_{k+2} \] Notice that the right-hand side uses the identity for Fibonacci numbers: \[ F_{k+2} = F_{k+1} + F_{k} \]Substitute this into the equation:\[ F_{k} F_{k+1} + F_{k+1}^2 = F_{k+1} (F_{k+1} + F_{k}) \]

Step 5: Simplify

Simplify the equation:\[ F_{k} F_{k+1} + F_{k+1}^2 = F_{k+1}^2 + F_{k+1} F_{k} \]Both sides are equal, confirming that the statement holds for \( n = k + 1 \). This completes the induction.

Key Concepts

Fibonacci sequence

The Fibonacci sequence is a famous number series that begins with 0 and 1, and each subsequent number in the series is the sum of the two preceding ones. It's defined as:

• \( F_0 = 0 \)
• \( F_1 = 1 \)
• \( F_n = F_{n-1} + F_{n-2} \) for \( n \geq 2 \)

This sequence appears in various aspects of mathematics and nature, such as the branching of trees, arrangement of leaves on a stem, and even in the pattern of various flowers.
Its simplicity and recursive nature make it an excellent subject for mathematical study. Each term after the first two is the result of adding the previous two terms together.
The Fibonacci sequence is a core concept of the given exercise, where the sequence is involved in proving a significant algebraic identity.

proof by induction

Proof by induction is a powerful mathematical technique used to establish the truth of an infinite number of statements. This method is akin to dominoes, where you show that each statement leads logically to the next.
Basically, induction involves two main steps:

• Base Case: Verify the statement is true for an initial value, typically for \( n = 1 \).
• Inductive Step: Show that if the statement holds true for \( n = k \), then it must be true for the next case \( n = k + 1 \).

By confirming these two steps, we prove that the statement is true for all natural numbers. In the given exercise, we use induction to prove an identity involving Fibonacci numbers by first demonstrating the base case and then using the inductive hypothesis to progress the argument to \( n = k + 1 \).
This structured approach helps establish the general truth from specific cases.

algebraic identities

Algebraic identities are equations that hold true for any values of their variables. They often express a fundamental truth or property shared across all numbers. In the exercise, we deal with the identity:\[F_{1}^{2}+F_{2}^{2}+\cdots+F_{n}^{2}=F_{n} F_{n+1}\]Identities like these allow us to perform complex manipulations or simplifications in algebraic expressions.
In the context of the given exercise, knowing that \( F_{k+2} = F_{k+1} + F_{k} \) is a key step in establishing the finale result as part of the inductive proof.
Understanding and applying these identities can make proving results like this more straightforward, as they often hold hidden symmetries or insights into how numerical elements relate to one another.

inductive hypothesis

The inductive hypothesis is a critical assumption used in mathematical induction. It's the logical step where we assume that a particular statement is true for some arbitrary natural number \( k \).
In essence, the hypothesis acts as a bridge to the next logical step.

• In this exercise, we assume the validity of \( F_{1}^{2}+F_{2}^{2}+\cdots+F_{k}^{2}=F_{k} F_{k+1} \).
• This assumption helps us build a case for \( n = k + 1 \).

With this assumption in place, the focus shifts to prove the next part of the chain—the inductive step.
By successfully demonstrating the truth of the statement for \( n = k + 1 \), the inductive hypothesis gets validated, allowing the statement to "carry forward" through all subsequent values.
It's a critical component without which the entire induction framework wouldn't function as effectively.