Question

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

Find ${\displaystyle \frac{{}^{a}n+1}{{}^{a}n}}$ 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:

$$a_3=a_2+a_1=1+1=2$$

$$a_4=a_3+a_2=2+1=3$$

$$a_5=a_4+a_3=3+2=5$$

$$a_6=a_5+a_4=5+3=8$$

$$\text{ For }n=1,\frac{a_{n+1}}{a_n}=\frac{a_2}{a_1}=\frac{1}{1}=1$$

$$\text{ For }n=2,\frac{a_{n+1}}{a_n}=\frac{a_3}{a_2}=\frac{2}{1}=2$$

$$\text{For }n=3,\frac{a_{n+1}}{a_n}=\frac{a_4}{a_3}=\frac{3}{2}$$

$$\text{ For }n=4,\frac{a_{n+1}}{a_n}=\frac{a_5}{a_4}=\frac{5}{3}$$

$$\text{ For }n=5,\frac{a_{n+1}}{a_n}=\frac{a_6}{a_5}=\frac{8}{5}$$