   Chapter 13.5, Problem 27E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# SKILLSProving a Factorization Show that x − y is a factor of x n − y n for all natural numbers n .[Hint: x k + 1 − y k + 1 = x k ( x − y ) + ( x k − y k ) y .]

To determine

To prove:

xy is a factor of xnyn for all natural numbers n.

Explanation

Approach:

Principal of Mathematical Induction:

Suppose P(n) is a statement depending on every natural number n and the following conditions are satisfied.

1.P(1) is true.

2. For every natural number k, if P(k) is true then P(k+1) is true.

Then P(n) is true for every natural number n.

Calculation:

Suppose P(n) denote the statement xy is a factor of xnyn.

Thus,

P(n)=xy is a factor of xnyn. …(1)

Step 1 Show that P(1) is true.

Substitute 1 for n in equation (1).

Clearly,

xy is a factor of x1y1.

This implies that P(1) is true.

Step 2 Assume that P(k) is true.

So, xkyk is divisible by xy

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