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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

Chapter 13.CR, Problem 72E, 70-72 Proof by InductionUse mathematical induction to show that the given statement is true. Formula 7 0 - 7 2 Proof by Induction Use mathematical induction to show that the given statement is true.

Formula for a Recursive Sequence A sequence is defined recursively by a n + 1 = 3 a n + 4 and a 1 = 4 . Show that a n = 2 3 n 2 for all natural numbers n .

To determine

To prove:

The statement an=23n2 for all natural numbers n for the sequence defined recursively as an+1=3an+4,a1=4.

Explanation

Given:

The sequence is given as,

an=23n2

When,

an+1=3an+4,a1=4

Approach:

Prove basic step for n=1. Assume the statement is true for some natural number n1 and show that the statement is true for n+1.

Calculation:

Denote the statement “an=23n2 for all natural numbers n for the sequence defined recursively as an+1=3an+4,a1=4” by P(n).

Step 1:

For, n=1

a1=4=2312

This proves Step 1.

Step 2:

Assume P(n) is true for some n1. This is called the induction hypothesis

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Chapter 13 Solutions

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