2580.61 – Sequence Formula

Consider the numbers FnF_n defined by the following formula:

Fn=(1+52 )n−(1−52 )n5F_n = \dfrac{ \left( \dfrac{1+\sqrt{5}}{2}\ \right)^n - \left( \dfrac{1-\sqrt{5}}{2}\ \right)^n}{ \sqrt{5}}

where nn may be any positive integer. Calculate F1F_1 through F5F_5. Notice something strange?

Solution

Did you get 1, 1, 2, 3, 5? Looks like the beginning of the Fibonacci sequence. Want to calculate a few more values to see if it continues? Doing it on a spreadsheet will save work.