Chapter 12: Q 25. (page 830)

In Problem, prove each statement.
$a - b$ is a factor of $a^{n} - b^{n}$

Short Answer

Expert verified

The given statement is shown.

Step by step solution

Step 1. Given information

The expression is $a + b$ .

Step 2. Prove that a-b is a factor of an-bn.

Put $n = 1$ in $a^{n} - b^{n}$

$a^{1} - b^{1} = a - b$

$a - b$ is a factor of $a - b$

Thus, the statement is true for $n = 1$

Assume that the statement is true for $n = k$

Thus, $a - b$ is a factor of $a^{k} - b^{k}$ .

Find whether $a - b$ is a factor of $a^{k + 1} - b^{k + 1}$

width="190">ak+1-bk+1=ak·a-bk·b

width="352">ak·a-bk·b+abk-abk=ak·a-abk+abk-bk·b=aak-bk+bk(a-b)

$a - b$ is a factor of $a^{k} - b^{k}$ . Thus, $a - b$ is a factor of $d^{k + 1} - b^{k + 1}$ .

Thus, the statement is true for $k + 1$

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Problem, prove each statement.
$a - b$ is a factor of $a^{n} - b^{n}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Prove that a-b is a factor of an-bn.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problem, prove each statement.a-bis a factor ofan-bn

Short Answer

Step by step solution

Step 1. Given information

Step 2. Prove that a-b is a factor of an-bn.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

In Problem, prove each statement.
$a - b$ is a factor of $a^{n} - b^{n}$