   Chapter 13.5, Problem 32E ### 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 Plus29-33 ■ Fibonacci Sequence F n denotes the nth term of the Fibonacci sequence discussed in section 13.1 . Use mathematical induction to prove the statement. F 1 + F 3 + ⋯ + F 2 n − 1 = F 2 n

To determine

To prove:

F1+F3++F2n1=F2n 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,

F1+F3++F2n1=F2n(2)

Step 1 Show that P(1) is true.

Substitute 1 for n in equation (2)

F1=F2

1=1

This implies that P(1) is true.

Step 2 Assume that P(k1) is true.

F1+F3++F2k3=F2k2(3)

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

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