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Chapter 8: Problem 42

Show that if \(a_{n}=a_{n-1}+a_{n-2}, \quad a_{0}=s\) and \(a_{1}=t\) where \(s\) and \(t\) are constants, then \(a_{n}=s f_{n-1}+t f_{n}\) for all positive integers \(n .\)

Short Answer

Expert verified
Use induction to show \(a_n = s f_{n-1} + t f_{n}\) holds for initial conditions and maintains via the recurrence relation.

Step by step solution

01

Understand the Problem

The problem gives a sequence defined by a recurrence relation: \(a_{n} = a_{n-1} + a_{n-2}\), with initial conditions \(a_{0} = s\) and \(a_{1} = t\). The goal is to show that \(a_{n} = s f_{n-1} + t f_{n}\), where \(f_n\) represents the Fibonacci sequence.
02

Recall Fibonacci Sequence

Recall the Fibonacci sequence definitions: \(f_{0} = 0\), \(f_{1} = 1\), and \(f_{n} = f_{n-1} + f_{n-2}\) for \(n \geq 2\).
03

Verify Initial Conditions

To ensure the formula holds true for the initial values, calculate:\(a_{0}\) and \(a_{1}\) using \(a_n = s f_{n-1} + t f_{n}\).\(a_{0} = s f_{-1} + t f_{0}\); since \(f_{-1} = 1\) (by extended definition) and \(f_{0} = 0\), we have \(a_{0} = s\), which matches the given condition. Similarly, calculate \(a_{1}:\)\(a_{1} = s f_{0} + t f_{1} = s(0) + t(1) = t\), confirming coherence with given conditions.
04

Assume the Formula for Inductive Step

Assume that the formula \(a_k = s f_{k-1} + t f_{k}\) holds true for all instances up to some integer \(k\). We will use this assumption to show that the formula still holds true for \(k + 1\).
05

Inductive Step

Show that \(a_{k+1}\) also fits the formula. By the recurrence relation,\(a_{k+1} = a_{k} + a_{k-1}\). Using induction assumption for \(a_k\) and \(a_{k-1}\):\(a_{k+1} = (s f_{k-1} + t f_{k}) + (s f_{k-2} + t f_{k-1})\)Regroup to combine like terms:\(a_{k+1} = s(f_{k-1} + f_{k-2}) + t(f_{k} + f_{k-1})\).
06

Simplify Using Fibonacci Properties

By definition of Fibonacci sequence:\(f_{k-1} + f_{k-2} = f_{k}\) and \(f_{k} + f_{k-1} = f_{k+1}\).Thus, calculating \(a_{k+1} = sf_{k} + t f_{k+1}\), which fits the prescribed formula.
07

Conclusion

As shown via induction hypothesis and initial verification, the expression \(a_n = s f_{n-1} + t f_{n}\) holds for all positive integers \(n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fibonacci sequence
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It starts with 0 and 1. Formally, it can be defined as: \(f_0 = 0\), \(f_1 = 1\), and for \(n \geq 2\), \(f_n = f_{n-1} + f_{n-2}\). This sequence has a lot of applications in various fields such as mathematics, computer science, and nature. The numbers in the Fibonacci sequence grow exponentially, meaning they get large very quickly. For example, the first few terms are: 0, 1, 1, 2, 3, 5, 8, and so on.
inductive reasoning
Inductive reasoning is a method of reasoning in which we seek to establish a general rule or conclusion from specific examples or patterns. In mathematics, this often involves showing that if a statement holds true for some initial case, and assuming it's true for some arbitrary case, we can prove it is then true for the next case. For example, in our exercise, we assumed the formula \(a_k = s f_{k-1} + t f_{k}\) is true for some number \k\ and then showed it must also be true for \k+1\. This allowed us to conclude that the formula is valid for all positive integers \.
initial conditions
Initial conditions are the starting values given in a recurrence relation. They are essential because they allow us to determine all subsequent terms in the sequence. In our exercise, the initial conditions are \(a_0 = s\) and \(a_1 = t\), where \s\ and \t\ are constants. These initial conditions provide the foundation from which all other terms in the sequence can be derived. Without these values, it would be impossible to commence the calculation of the sequence.
sequence proof
A sequence proof is a logical demonstration that shows a given property holds for every element in a sequence. In our exercise, we needed to prove that \(a_n = s f_{n-1} + t f_{n}\) for all positive integers \. To do this, we used both the initial conditions and inductive reasoning. We first verified the formula for the initial values \(n = 0\) and \(n = 1\). Then, by assuming it holds for some \k\ and demonstrating it must also hold for \k+1\, we completed our proof. This step-by-step logical process ensures that the formula is valid throughout the entire sequence.

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