   Chapter 11.2, Problem 88E

Chapter
Section
Textbook Problem

The Fibonacci sequence was defined in Section 11.1 by the equations f 1 = 1 , f 2 = 1 , f n = f n − 1 + f n − 2 n ≥ 3 Show that each of the following statements is true.(a) 1 f n − 1 f n + 1 = 1 f n − 1 f n − 1 f n f n + 1 (b) ∑ n = 2 ∞ 1 f n − 1 f n + 1 = 1 (c) ∑ n = 2 ∞ f n f n − 1 f n + 1 = 2

To determine

Part (a):

To show:

The statement 1fn-1fn+1=1fn-1fn-1fnfn+1 is true

Explanation

1) Concept:

The Fibonacci sequence is defined recursively by the conditions

f1=1, f2=1, fn=fn-1+fn-2       n3

2) Given:

1fn-1fn+1=1fn-1fn-1fnfn+1

3) Calculation:

Consider RHS of  the given statement

1fn-1fn-1fnfn+1=fnfn+1-fn-1fnfn2fn-1fn+1

Factor out fn from the numerator

=fn(fn+1-fn-1)fn2fn-1fn

To determine

Part (b):

To show:

The statement

n=21fn-1fn+1=1 is true

To determine

Part (c):

To show:

The statement n=2fnfn-1fn+1=2 is true

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