Question

The Fibonacci sequence is defined by
a₁ = 1 = a₂, a_n = a_{n − 1} + a_{n − 2} for n > 2

Find $\frac{{}^{a}n+1}{{}^{a}n}$ for n = 1, 2, 3, 4, 5.

Answer 1

a₁ = 1 = a₂, a_n = a_{n − 1} + a_{n − 2} for n > 2
Then, we have:
$$\begin{gathered} a_3=a_2+a_1=1+1=2 \\ {a}_4=a_3+a_2=2+1=3 \\ {a}_5=a_4+a_3=3+2=5 \\ {a}_6=a_5+a_4=5+3=8 \\ \mathrm{For}n=1,\frac{a_{n+1}}{a_n}=\frac{a_2}{a_1}=\frac{1}{1}=1 \\ \mathrm{For}n=2,\frac{a_{n+1}}{a_n}=\frac{a_3}{a_2}=\frac{2}{1}=2 \\ \mathrm{For}n=3,\frac{a_{n+1}}{a_n}=\frac{a_4}{a_3}=\frac{3}{2} \\ \mathrm{For}n=4,\frac{a_{n+1}}{a_n}=\frac{a_5}{a_4}=\frac{5}{3} \\ \mathrm{For}n=5,\frac{a_{n+1}}{a_n}=\frac{a_6}{a_5}=\frac{8}{5} \end{gathered}$$