In Pursuit of an Nth Term Formula

So there already exists an nth term formula for the Fibonacci sequence… but it sucks. It’s more of a definition than a practical formula.

Let’s say I wanted to find the 14938593th Fibonacci number. Using this formula, I would have to first find the 14938592nd and the 14938591st Fibonacci number before I could find their successor. Doesn’t that sound fun?

In order to play with the Fibonacci sequence in trigonometric functions, we’d need a more pragmatic formula for the nth term of the sequence. The following was presented to me:

Although this formula looks tempting, I have yet to find a peer-reviewed source that says it is legitimate. So I undertook the task of proving or disproving it:

So far, so good. Stay tuned for the second half of the proof!