Problem 32 - 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. $$ F_{1}+F_{3}+\cdots+F_{2 n-1}=F_{2 n} $$

Step-by-Step Solution

Answer

By induction, \( F_1 + F_3 + \cdots + F_{2n-1} = F_{2n} \) is true for all \( n \).

Step 1: Introduction to the Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It starts from 0 and 1. That is, \( F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, \) and so forth.

Step 2: Statement of Problem for Induction

We need to prove by mathematical induction that the sum of odd-indexed Fibonacci terms up to \( F_{2n-1} \) equals the even-indexed term \( F_{2n} \), i.e., \( F_1 + F_3 + \cdots + F_{2n-1} = F_{2n} \) for any positive integer \( n \).

Step 3: Base Case

First, verify the statement for \( n = 1 \). The expression becomes \( F_1 = F_2 \). Since \( F_1 = 1 \) and \( F_2 = 1 \), the base case holds true: \( 1 = 1 \).

Step 4: Inductive Hypothesis

Assume the statement is true for some integer \( k \), i.e., \( F_1 + F_3 + \cdots + F_{2k-1} = F_{2k} \). We need to show it holds for \( k+1 \), that is, \( F_1 + F_3 + \cdots + F_{2k-1} + F_{2k+1} = F_{2k+2} \).

Step 5: Inductive Step

Using the inductive hypothesis, assume \( F_1 + F_3 + \cdots + F_{2k-1} = F_{2k} \). Then the sum \( F_1 + F_3 + \cdots + F_{2k-1} + F_{2k+1} = F_{2k} + F_{2k+1} \). According to the properties of Fibonacci numbers, \( F_{2k} + F_{2k+1} = F_{2k+2} \). Therefore, the statement holds for \( k+1 \).

Step 6: Conclusion by Induction

Since both the base case holds and the inductive step shows the statement is true for \( n = k+1 \) assuming it is true for \( n = k \), by the principle of mathematical induction, the statement is proven for all positive integers \( n \): \( F_1 + F_3 + \cdots + F_{2n-1} = F_{2n} \).

Key Concepts

Fibonacci sequence

The Fibonacci sequence is a fascinating series that appears in various branches of mathematics and is observed in different natural phenomena. The sequence begins with the numbers 0 and 1. From there, each subsequent number is the sum of the two preceding numbers: for instance, 0+1 gives 1, 1+1 equals 2, 1+2 results in 3, and so on.

• Initial terms: 0, 1, 1, 2, 3, 5, 8, 13, ...
• Rule: Each term is the sum of the two preceding ones.

In mathematical form:\[ F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5, \dots \]
This sequence grows incredibly fast, and it appears in a surprising number of fields, including computer science, biology (for instance, in the arrangement of leaves or flowers), and art.

inductive hypothesis

The inductive hypothesis is a crucial element when using mathematical induction, which is a method of proof in mathematics. To use induction, you first assume that a statement is true for a particular case, usually denoted as \( k \).

• This assumption is what we call the "inductive hypothesis."
• For our problem, we assume that for some integer \( k \), \( F_1 + F_3 + \cdots + F_{2k-1} = F_{2k} \) holds true.
• This assumption forms the basis from which we prove the next step.

By making this assumption, you prepare a foundation to help prove that if the statement holds for \( k \), it should also be true for \( k+1 \). This is an essential step in inductive proofs, providing a bridge between the known and the unknown.

inductive step

In the process of mathematical induction, the inductive step is where you prove that if a statement holds true for a certain case \( k \) (as per your inductive hypothesis), it will also hold true for the next case \( k+1 \).

• The inductive step uses the assumption from the inductive hypothesis.
• So, we assume that \( F_1 + F_3 + \cdots + F_{2k-1} = F_{2k} \).
• Then, we need to prove \( F_1 + F_3 + \cdots + F_{2k-1} + F_{2k+1} = F_{2k+2} \).
• Using the property of the Fibonacci sequence: \( F_{2k+2} = F_{2k} + F_{2k+1} \).

Once this step is successfully executed, it indicates that the statement, with its starting point of the base case, can continue to extend to any positive integer.

base case

The base case in mathematical induction is where you begin, showing the statement is true for the initial value. It acts like the anchor to the whole inductive reasoning process, ensuring the entire "chain reaction" can safely start.

• For our sequence, we check for \( n=1 \).
• In our specific problem, the base case checks if \( F_1 = F_2 \).
• This turns to \( 1 = 1 \), which is true.

This initial step is critical as it establishes that the statement holds when \( n \) is the smallest possible positive integer. Without a solid base case, the inductive proof could not logically proceed or verify any subsequent cases.