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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

1 5 2 4 Proving a Statement Use mathematical induction to show that the given statement is true.

( n + 1 ) 2 < 2 n 2 for all natural numbers n 3 .

To determine

To prove:

(n+1)2<2n2 for all natural numbers n3.

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:

Suppose,

P(n)=(n+1)2<2n2(1)

For all n3

Step 1 Show that P(3) is true.

Substitute 3 for n in equation (1).

(3+1)2<2(3)216<18,

which is true.

Thus, P(3) is true.

Step 2 Assume that P(k) is true.

Substitute k for n in equation (1).

(k+1)2<2k2(2)

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

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