3
$\begingroup$

Define $\ a_n=\frac{F_{n+1}}{F_n},n>1$ where $F_n$ is a member of a Fibonacci sequence.

  • a) Write the first 10 terms of $a_n$

  • b) Show that $a_n= \frac1{a_n-1}$

I was able to solve the a) subquestion, however, I cannot solve b). Any help and/or hints would be greatly appreciated.

  • 0
    $$\dfrac1{\dfrac{F_{n+1}}{F_n}-1}=\dfrac{F_n}{F_{n-1}}$$ We need $$\dfrac{F_{n+1}}{F_n}=\dfrac{F_n}{F_{n-1}}\iff F^2_n=F_{n-1}F_{n+1}$$ http://math.stackexchange.com/questions/523925/induction-proof-on-fibonacci-sequence-fn-1-cdot-fn1-fn2-1n – 2017-02-02

1 Answers 1

1

We have the Fibonacci sequence as $$1,1,2,3,5,8,13,21,34,55,...$$ then $$a_1=1, a_2=2, a_3=3/2, a_4=5/3, a_5=8/5,a_6=13/8$$, $$a_7=21/13, a_8=34/21, a_9=55/34, a_{10}=89/55$$ Now as we know about Fibonacci sequence; we have $F_{n+1}=F_n+F_{n-1}$. Then by your assumption $a_n=\frac{F_{n+1}}{F_n}$ we get $a_nF_n=F_{n+1}=F_n+F_{n-1}$ then $$a_nF_n-F_n=F_{n-1}\Rightarrow F_n(a_n-1)=F_{n-1}$$ So we have $$a_n-1=\frac{F_ {n-1}}{F_n}=\frac{1}{a_{n-1}}$$