   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

(a)

To determine

To show:  The statement 1fn1fn+1=1fn1fn1fnfn+1 is true.

Explanation

Given:

The Fibonacci sequence was defined by the equations f1=1, f2=1, fn=fn1+fn2 .

Proof:

Obtain 1fn1fn1fnfn+1 .

1fn1fn1fnfn+1=fnfn+1fn1fn(fn1fn)(fnfn+1)=fn(fn+1fn1)

(b)

To determine

To show:  The series n=21fn1fn+1=1 .

(c)

To determine

To show: The series n=2fnfn1fn+1=2 .

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