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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

2 9 3 3 Fibonacci Sequence F n denotes the n th term of the Fibonacci sequence discussed in Section 13.1 . Use mathematical induction to prove the statement.

F 1 + F 2 + F 3 + + F n = F n + 2 1

Section 1 3 . 1 .

EXAMPLE 4 The Fibonacci Sequence

Find the first 11 terms of the sequence defined recursively by F 1 = 1 , F 2 = 1 , and F n = F n 1 + F n 2 .

To determine

To prove:

F1+F2+F3++Fn=Fn+21

Explanation

Given:

Fn denotes the nth term of the Fibonacci sequence.

The Fibonacci sequence Fn is recursively defined by Fn=Fn1+Fn2(1)

Here, F1=1

F2=1

Approach:

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 all natural numbers n.

Consider,

F1+F2+F3++Fn=i=1nFi

Therefore,

i=1nFi=Fn+21(2)

Calculation:

Let P(n) denote the Fibonacci sequence i=1nFi=Fn+21.

Step 1 Show that P(1) is true.

Substitute 3 for n, 1 for F1, and 1 for F2 in equation (1).

F3=F31+F32=F2+F1=1+1=2

Substitute 1 for i in equation (2).

F1=F1+21=F31=21=1 (Because F3=2)

This implies that P(1) is true

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Chapter 13 Solutions

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Sect-13.1 P-11ESect-13.1 P-12ESect-13.1 P-13ESect-13.1 P-14ESect-13.1 P-15ESect-13.1 P-16ESect-13.1 P-17ESect-13.1 P-18ESect-13.1 P-19ESect-13.1 P-20ESect-13.1 P-21ESect-13.1 P-22ESect-13.1 P-23ESect-13.1 P-24ESect-13.1 P-25ESect-13.1 P-26ESect-13.1 P-27ESect-13.1 P-28ESect-13.1 P-29ESect-13.1 P-30ESect-13.1 P-31ESect-13.1 P-32ESect-13.1 P-33ESect-13.1 P-34ESect-13.1 P-35ESect-13.1 P-36ESect-13.1 P-37ESect-13.1 P-38ESect-13.1 P-39ESect-13.1 P-40ESect-13.1 P-41ESect-13.1 P-42ESect-13.1 P-43ESect-13.1 P-44ESect-13.1 P-45ESect-13.1 P-46ESect-13.1 P-47ESect-13.1 P-48ESect-13.1 P-49ESect-13.1 P-50ESect-13.1 P-51ESect-13.1 P-52ESect-13.1 P-53ESect-13.1 P-54ESect-13.1 P-55ESect-13.1 P-56ESect-13.1 P-57ESect-13.1 P-58ESect-13.1 P-59ESect-13.1 P-60ESect-13.1 P-61ESect-13.1 P-62ESect-13.1 P-63ESect-13.1 P-64ESect-13.1 P-65ESect-13.1 P-66ESect-13.1 P-67ESect-13.1 P-68ESect-13.1 P-69ESect-13.1 P-70ESect-13.1 P-71ESect-13.1 P-72ESect-13.1 P-73ESect-13.1 P-74ESect-13.1 P-75ESect-13.1 P-76ESect-13.1 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