Question

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

Find `(""^an +1)/(""^an")` 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:

\[a_3 = a_2 + a_1 = 1 + 1 = 2\]

\[ a_4 = a_3 + a_2 = 2 + 1 = 3\]

\[ a_5 = a_4 + a_3 = 3 + 2 = 5\]

\[ a_6 = a_5 + a_4 = 5 + 3 = 8\]

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

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

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

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

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