/*! 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} Q 20. In Problem, use the Principle of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Q 20. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

n3+2n is divisible by3

Short Answer

Expert verified

The given statement is shown.

Step by step solution

01

Step 1. Given information

The given expression isn3+2n

02

Step 2. Show that the given statement is correct.

First we consider the statement for n=1

Because 13+2.1=3is divisible by 3 , the statement is true for width="37">n=1

Now we consider that the statement holds for some k

We need to show that (k+1)3+2(k+1)is divisible by 3

width="305">(k+1)3+2(k+1)=k3+3k2+3k+1+2k+2=k3+2k+3k2+3k+3=k3+2k+3k2+k+1

Since 3k2+k+1is divisible by 3 and k3+2kis divisible by 3 , it follows that (k+1)3+2(k+1)is divisible by 3.

As a result statement 1 is true for all natural numbersn

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!

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

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

a1=1;an=n-an-1

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

a1=3;an=4-an-1

In Problems 71-82, find the sum of each sequence.

鈭�(-k=124k)

In Problems 61鈥�70, express each sum using summation notation.

3+322+333+.....+3nn

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

鈭�k=1鈭�123k-1
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

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.