/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Use mathematical induction in Ex... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 35

Use mathematical induction in Exercises \(31-37\) to prove divisibility facts. Prove that \(n^{2}-1\) is divisible by 8 whenever \(n\) is an odd positive integer.

Short Answer

Expert verified
By mathematical induction, n^2 - 1 is divisible by 8 for all odd positive integers n.

Step by step solution

01

- Base Case

First, verify the statement for the base case when n is 1. For n = 1, evaluate the expression: n^2 - 1 = 1^2 - 1 = 0 Clearly, 0 is divisible by 8, so the base case holds.
02

- Inductive Hypothesis

Assume that for an odd positive integer k, the statement is true. That is, k^2 - 1 is divisible by 8.
03

- Inductive Step

To prove the statement for k + 2 (the next odd integer after k), evaluate the expression: (k + 2)^2 - 1 = k^2 + 4k + 4 - 1 Simplify to get: k^2 + 4k + 3.
04

- Verification

Notice that k^2 - 1 is divisible by 8 by the inductive hypothesis. We need to show that the entire expression k^2 + 4k + 3 - (k^2 - 1) is divisible by 8. Simplifying, we get: 4k + 4.
05

- Simplified Expression

It follows that 4k + 4 = 4(k + 1). Since k is an odd integer, k + 1 is even, and thus, 4(k + 1) is divisible by 8.
06

Conclusion

Since the base case holds and the inductive step has been proven, by mathematical induction, the statement n^2 - 1 is divisible by 8 for all odd positive integers n.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

divisibility
Divisibility is a fundamental concept in mathematics. It means one number can be evenly divided by another without leaving a remainder. When we say that a number, say 鈥榓,鈥� is divisible by another number 鈥榖,鈥� it means there exists an integer 鈥榢鈥� such that a = kb. In more formal terms, we say that b divides a or b is a divisor of a. For instance, if we say the number 15 is divisible by 5, we mean that 15 = 5 * 3, where 3 is the integer. In the context of our exercise, we proved that for any odd positive integer n, the expression n^2 - 1 is divisible by 8. Divisibility by 8 means that when n^2 - 1 is divided by 8, the remainder is 0, indicating an exact division.
base case
The base case is a crucial part of a mathematical induction proof. It is the first step where we verify that our statement holds true for the initial value of the variable, typically the smallest number in the set. In our proof, we started with n = 1, the smallest odd positive integer. We evaluated the expression n^2 - 1 for n = 1 and found that 1^2 - 1 = 0. Since 0 is divisible by 8, we concluded that the base case holds true. The base case establishes a solid foundation that our assumption in the inductive hypothesis can rely on.
inductive hypothesis
The inductive hypothesis is the second step in the process of mathematical induction. In this step, we assume that our statement is true for some arbitrary odd positive integer k. This assumption forms the basis for proving the statement for the next value, typically k + 2 in the case of odd integers. In our exercise, we took an odd positive integer k and assumed that k^2 - 1 is divisible by 8. This hypothesis allows us to link the truth of the statement for k to the truth of the statement for k + 2, moving forward incrementally.
inductive step
The inductive step is the third and final step in a mathematical induction proof. Here, we demonstrate that if the statement holds for k, it also holds for k + 2, the next odd integer. We started with the expression for k + 2: (k + 2)^2 - 1. Simplifying, we obtained k^2 + 4k + 3. To show divisibility by 8, we subtracted k^2 - 1 (known to be divisible by 8) from k^2 + 4k + 3, giving us 4k + 4. Factoring 4 out, we got 4(k + 1). Since k is odd, k + 1 is even, meaning 4(k + 1) must be divisible by 8. Thus, the statement holds for k + 2, completing the inductive step and the proof.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider this variation of the game of Nim. The game begins with \(n\) matches. Two players take turns removing matches, one, two, or three at a time. The player removing the last match loses. Use strong induction to show that if each player plays the best strategy possible, the first player wins if \(n=4 j, 4 j+2,\) or \(4 j+3\) for some nonnega- tive integer \(j\) and the second player wins in the remaining case when \(n=4 j+1\) for some nonnegative integer \(j .\)

Give iterative and recursive algorithms for finding the \(n\) th term of the sequence defined by \(a_{0}=1, a_{1}=3, a_{2}=5\) and \(a_{n}=a_{n-1} \cdot a_{n-2}^{2} \cdot a_{n-3}^{3} .\) Which is more efficient?

Show that \(\left[\left(p_{1} \rightarrow p_{2}\right) \wedge\left(p_{2} \rightarrow p_{3}\right) \wedge \cdots \wedge\left(p_{n-1} \rightarrow p_{n}\right)\right]\) \(\quad \rightarrow\left[\left(p_{1} \wedge p_{2} \wedge \cdots \wedge p_{n-1}\right) \rightarrow p_{n}\right]\) is a tautology whenever \(p_{1}, p_{2}, \ldots, p_{n}\) are propositions, where \(n \geq 2\)

Suppose that \(A\) and \(B\) are square matrices with the property \(A B=B A .\) Show that \(A B^{n}=B^{n} A\) for every positive integer \(n .\)

Assume that a chocolate bar consists of \(n\) squares ar- ranged in a rectangular pattern. The entire bar, or any smaller rectangular piece of the bar, can be broken along a vertical or a horizontal line separating the squares. Assuming that only one piece can be broken at a time, determine how many breaks you must successively make to break the bar into \(n\) separate squares. Use strong in- duction to prove your answer.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.