   Chapter 13.5, Problem 30E ### 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

# 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 SequenceFind 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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