Answer: The Fibonacci sequence is defined recursively by where and (a) Show that (b)

Chapter 9, Problem 135

(choose chapter or problem)

The Fibonacci sequence is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}\), where \(a_{1}=1\) and \(a_{2}=1\).

(a) Show that \(\frac{1}{a_{n+1} a_{n+3}}=\frac{1}{a_{n+1} a_{n+2}}-\frac{1}{a_{n+2} a_{n+3}}\).

(b) Show that \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).

Text Transcription:

a_n+2=a_n+a_n+1

a_1=1

a_2=1

frac1a_n+1a_n+3=frac1a_n+1a_n+2-frac1a_n+2a_n+3

sum_n=0^inftyfrac1a_n+1a_n+3=1