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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 Plus

29-33 ■ Fibonacci Sequence F n denotes the nth term of the Fibonacci sequence discussed in Section 13.1 . Use mathematical induction to prove the statement.

For all n 2 ,

[ 1 1 1 0 ] n = [ F n + 1 F n F n F n 1 ]

To determine

To prove:

For all n2, [1110]n=[Fn+1FnFnFn1] by mathematical induction

Explanation

Given:

Fn denote the nth term of the Fibonacci sequence.

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

Here, F0=0

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(k1) is true then P(k) is true.

Then P(n) is true for all natural numbers n.

Calculation:

Let P(n) denote the Fibonacci sequence,

[1110]n=[Fn+1FnFnFn1](2)

Step 1 Show that P(2) is true.

Substitute 2 for n in equation (2)

[1110]2=[F3F2F2F1][2110]=[2110]

This implies that P(1) is true.

Step 2 Assume that P(k) is true.

[1110]k=[Fk+1FkFkFk1](3)

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

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Chapter 13 Solutions

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