   Chapter 13.5, Problem 19E ### 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

# SKILLS 1 5 − 2 4 Proving a Statement Use mathematical induction to show that the given statement is true. 8 n − 3 n is divisible by 5 for all natural numbers n .

To determine

To prove:

8n3n is divisible by 5 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 all natural numbers n.

Calculation:

Let P(n) denote the expression 8n3n.

P(n)=8n3n(1)

Step 1 Show that P(1) is true.

Substitute 1 for n in equation (1).

8131=5, which is divisible by 5.

This implies that P(1) is true.

Step 2 Assume that P(k) is true.

Substitute k for n in equation (1).

8k3k=5S,

Here, S belongs to integers.

8k=5S+3k(2)

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

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