Fibonacci Sequence The Fibonacci sequence is defined recursively by where and (a) Show
Chapter 9, Problem 99(choose chapter or problem)
Fibonacci Sequence 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
1/a_n+1 a_n+3=1/a_n+1 a_n+2 - 1/a_n+1 a_n+3
Sum_n=0^infinity 1/a_n+1 a_n+3 = 1
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