Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 11 - Sequences; Induction; the Binomial Theorem - Section 11.4 Mathematical Induction - 11.4 Assess Your Understanding - Page 850: 25

Answer

See below.

Work Step by Step

1. Test for $n=1$, $a-b$ is a factor of $a^1-b^1$, it is true. 2. Assume the statement is true for $n=k$, that is $a-b$ is a factor of $a^k-b^k$, 3. For $n=k+1$, use the given hint, we have $a^{k+1}-b^{k+1}=a(a^k-b^k)+b^k(a-b)$ which contains two part with each part containing a factor of $(a-b)$, thus it is true for $n=k+1$, 4. We can conclude that the statement is true for any integer $n$.
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