Mathematico | Mathematico

The most famous recursive sequence is probably the Fibonacci sequence:

\begin{aligned} f_{1} & = 1 \\ f_{2} & = 1 \\ f_{n + 1} & = f_{n} + f_{n - 1} \end{aligned}

Work out $f_{3}, f_{4}, f_{5}$ and $f_{6}$
Define a new sequence $g_{n}$ by
$\begin{aligned} g_{1} & = 1 \\ g_{n + 1} & = 1 + \frac{1}{g_{n}} \end{aligned}$
Find $g_{2}, g_{3}, g_{4}$ and $g_{5}$
What is the relationship between these two sequences? Can you explain why?

To get started with (a), let $n = 1$ to get $f_{3} = f_{2} + f_{1}$ .

Then let $n = 2$ to get $f_{4} = f_{3} + f_{2} .$

Keep going until you get to $f_{6} .$