Problem 30 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The statement is proven by induction; it holds for all positive integers \( n \).

Step 1: Base Case

To begin mathematical induction, first confirm the statement holds for the smallest value of \( n \), which is \( n = 1 \).\( F_1 = 1 \) and \( F_{1+2} - 1 = F_3 - 1 = 2 - 1 = 1 \).Therefore, the base case holds as \( F_1 = F_3 - 1 \).

Step 2: Induction Hypothesis

Assume the statement holds for some arbitrary positive integer \( k \). This means\[ F_1 + F_2 + F_3 + \cdots + F_k = F_{k+2} - 1. \]

Step 3: Inductive Step

We must show that if the statement holds for \( n = k \), it also holds for \( n = k + 1 \).Start from the assumption:\[ F_1 + F_2 + F_3 + \cdots + F_k + F_{k+1} = F_{k+2} - 1 + F_{k+1}. \]

Step 4: Simplify Right Side

Use the Fibonacci property that \( F_{n+2} = F_{n+1} + F_n \), which implies:\[ F_{k+2} + F_{k+1} = F_{k+3}. \]So,\[ F_{k+2} - 1 + F_{k+1} = F_{k+3} - 1. \]

Step 5: Conclusion by Induction

Since both sides simplify to the same expression, we can conclude that\[ F_1 + F_2 + F_3 + \cdots + F_{k+1} = F_{k+3} - 1. \]Therefore, by induction, the statement is true for all positive integers \( n \).

Key Concepts

Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each term is the sum of the two preceding ones. It starts with 0 and 1: 0, 1, 1, 2, 3, 5, 8, and so on.
In mathematical terms, the sequence is defined by the recurrence relation,
\( F_n = F_{n-1} + F_{n-2} \) with seed values \( F_1 = 1 \) and \( F_2 = 1 \).
This sequence appears in various natural phenomena, from the branching of trees to the arrangement of leaves on a stem.
One remarkable property of the Fibonacci Sequence is its ability to model growth patterns over time. This characteristic makes it a fundamental concept in mathematics and nature.

Base Case

The Base Case is the initial step in a mathematical induction proof. It demonstrates that the statement you want to prove holds true for the first element of the sequence.
In this context, the base case involves proving that the equation \( F_1 = F_3 - 1 \) holds true.
This involves substituting the first term of the Fibonacci Sequence into the equation to verify it:

• \( F_1 = 1 \)
• \( F_3 - 1 = 2 - 1 = 1 \)

A successful base case confirms that the statement is indeed true for the smallest possible value.
This sets the foundation for moving onto the induction hypothesis.

Inductive Step

The Inductive Step is a crucial part of mathematical induction. It shows that if a statement holds for one number, it also holds for the next one in the sequence.
To begin, assume the statement is true for some arbitrary positive integer \( k \).
In this exercise, the assumption is:

• \( F_1 + F_2 + F_3 + \cdots + F_k = F_{k+2} - 1 \)

This is known as the induction hypothesis.
Next, you need to prove it holds for \( n = k + 1 \).
We start by adding \( F_{k+1} \) to both sides:

• \( F_1 + F_2 + F_3 + \cdots + F_{k+1} = F_{k+2} - 1 + F_{k+1} \)

This uses the property of the Fibonacci Sequence that \( F_{n+2} = F_{n+1} + F_n \).
Showing that both sides simplify to \( F_{k+3} - 1 \) completes the inductive step.

Mathematical Proof

A Mathematical Proof is a logical argument that verifies the truth of a given statement. By using accepted mathematical principles, proofs provide certainty in mathematics.
An induction proof, in particular, involves two main steps:

• Base Case: Showing the statement holds for the first element.
• Inductive Step: Proving if it holds for one element, it holds for the next one.

This method allows us to assert the validity of a statement for an infinite number of elements.
In the provided exercise, the sequence starts with verifying the smallest value, then uses the inductive step to generalize the statement to all positive integers \( n \).
The proof uses the properties of the Fibonacci Sequence to show this conversion.
By conducting these steps thoroughly, the original mathematical statement is convincingly verified.