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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

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

SKILLS

Proving a Factorization Show that x + y is a factor of x 2 n 1 + y 2 n 1 for all natural numbers n .

To determine

To prove:

x+y is a factor of x2n1+y2n1 for all natural numbers n.

Explanation

Approach:

Principal of Mathematical Induction:

Let P(n) be a statement depending on every natural number n. Suppose that 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 x+y is a factor of x2n1+y2n1.

Thus,

P(n)= x+y is a factor of x2n1+y2n1. …(1)

Step 1 Show that P(1) is true

Substitute 1 for n in equation (1).

Clearly,

x+y is a factor of x21+y21=x+y.

This implies that P(1) is true.

Step 2 Assume that P(k) is true.

So, x2k1+y2k1 is divisible by x+y.

Step 3 Show that it is true for P(k+1).

Substitute k+1 for n in equation (1).

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Chapter 13 Solutions

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