   Chapter 2.2, Problem 19E

Chapter
Section
Textbook Problem

Let x   and   y be integers, and let m   and   n be positive integers. Use induction to prove the statements in Exercises 18 − 23 . ( The definitions of x n and n x are given before Theorem 2.5 in Section 2.1 .) x m ⋅ x n = x m + n

To determine

To prove: That xmxn=xm+n is true for all n by using mathematical induction.

Explanation

Formula Used:

Considering the given statement is Pn for all integers n,

a. if Pn is true for n=1

b. if the truth of Pk always implies that Pk+1 is true, then the statement Pn is true for all positive integers n.

Proof:

For each positive integer n, let Pn be the statement

xmxn=xm+n

When n=1, the value of the left side is

xmx1=xm+1

And the value of right side is xm+1.

Thus P1 is true.

Assume now that Pk is true. That is, assume that the equation

xmxn=xm+n is true.

With this assumption made, prove that Pk+1 is true

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

let f(x) = x 1, g(x) = x+1, and h(x) = 2x3 1. Find the rule for each function. 12. gf

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Expand each expression in Exercises 122. (y1y)2

Finite Mathematics and Applied Calculus (MindTap Course List)

Evaluate the limit, if it exists. limx4x2+95x+4

Single Variable Calculus: Early Transcendentals, Volume I

Evaluate the integral. 65. sin2x1+cos4xdx

Single Variable Calculus: Early Transcendentals

True or False: If h'(x) = k(x), then k(x) is an antiderivative of h(x).

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 