Problem 15 - Algebra 2

Question

NATURE The terms of the Fibonacci sequence are found in many places in nature. The number of spirals of seeds in sunflowers are Fibonacci numbers, as are the number of spirals of scales on a pinecone. The Fibonacci sequence begins \(1,1,2,3,5,8, \ldots\) Each element after the first two is found by adding the previous two terms. If \(f_{n}\) stands for the \(n\) th Fibonacci number, prove that \(f_{1}+f_{2}+\ldots+f_{n}=f_{n+2}-1\)

Step-by-Step Solution

Answer

The formula holds by mathematical induction for all positive integers n.

Step 1: Understanding the Problem

The problem asks us to prove that the sum of the first n Fibonacci numbers is equal to the (n+2)th Fibonacci number minus 1. We need to show that: \(f_{1} + f_{2} + \ldots + f_{n} = f_{n+2} - 1\).

Step 2: Inductive Proof Base Case

First, verify the base case for induction, where n = 1. The left side is \(f_1 = 1\). The right side is \(f_{3} - 1 = f_3 - 1 = 2 - 1 = 1\). Both sides are equal, satisfying the base case.

Step 3: Inductive Step Hypothesis

Assume that the formula is true for some arbitrary positive integer k, i.e., \(f_{1} + f_{2} + \ldots + f_{k} = f_{k+2} - 1\).

Step 4: Inductive Step Conclusion

Show that the formula holds for \(n = k + 1\): Start from: \(f_{1} + f_{2} + \ldots + f_{k} + f_{k+1}\). According to the hypothesis: \(f_{1} + f_{2} + \ldots + f_{k} = f_{k+2} - 1\). So: \(f_{1} + f_{2} + \ldots + f_{k} + f_{k+1} = f_{k+2} - 1 + f_{k+1}\).Since \(f_{k+3} = f_{k+2} + f_{k+1}\),We have that \(f_{1} + f_{2} + \ldots + f_{k} + f_{k+1} = f_{k+3} - 1\).

Step 5: Conclusion of Inductive Proof

Both the base case and the inductive step show that the statement is true for all positive integers n. Thus, the original statement \(\sum_{i=1}^{n}f_{i} = f_{n+2} - 1\) is proven.

Key Concepts

mathematical induction

Mathematical induction is a powerful tool used in mathematics to prove statements that are true for an infinite number of cases. It can be thought of as a domino effect. If you can prove that the first domino in a long line will fall and that whenever any one domino falls, the next one will also go down, then all the dominoes will eventually fall. Induction involves two main steps:

• Base Case: Show that a statement holds for the initial value (often starting with the first term).
• Inductive Step: Assume that the statement holds for an arbitrary case (usually for an integer \( k \)) and then prove it is true for the next case (for \( k+1 \)).

In the context of the Fibonacci sequence problem, induction helped establish that if the sum of up to the \( k \)-th Fibonacci number works, it must also hold true for the \( (k+1) \)-th number. This step-by-step method confirms that the formula is valid for all positive integers.

sum of Fibonacci numbers

The Fibonacci sequence is a fascinating series where each number is the sum of the two preceding ones, often starting with 1 and 1. The sequence looks like this: 1, 1, 2, 3, 5, 8, and so on. When you add up the sequence from 1 to any point \( n \), you get what's known as the sum of Fibonacci numbers.
In the exercise, we considered the sum \( f_1 + f_2 + \ldots + f_n \), and proved that it equals \( f_{n+2} - 1 \). This relationship beautifully links the sum of the series to a particular Fibonacci number itself. So, for any given \( n \), knowing the first \( n \) Fibonacci numbers, you can effortlessly calculate their sum using the formula derived. It simplifies calculations by turning a long addition problem into a simple lookup and subtraction task.

proof by induction

Proof by induction is a streamlined approach to confirm an infinite number of cases with just a few logical steps. It is deeply connected with the concept of recursion since it relies on assuming a truth for a small problem size and then using it to solve larger problems.
To prove the given formula for the sum of Fibonacci numbers by induction, we:

• Started by verifying that the formula holds when \( n = 1 \). This was our base case.
• Then, we assumed the formula was true for some \( n = k \), our inductive hypothesis.
• Using this assumption, we showed it also had to be true for \( n = k + 1 \), therefore completing the proof.

Inductive proofs can seem tricky at first, but they are similar to checking that a machine functions correctly for the smallest task before ensuring it works for larger tasks. In this way, induction manages complexity by focusing on one step at a time, making it versatile for proving statements involving sequences, as seen with the Fibonacci numbers.