A stairway has 100 steps. You can climb it by one step at a time, or by two steps. How many different ways to ascend this stairway exist?

(In reply to A new approach with no proof by TomM)

I think I can justify this trend...

Generalize the stairway to n steps.

Think about the last step you will take on the stairs. It is either one-step or two-step. Which means you are either standing at n-1 or at n-2.

So the number of ways to ascend n steps is equal to the number of ways to get to step n-1 plus the number of ways to get to step n-2. This is exactly the Fibonacci sequence.

Posted by levik on 2002-09-05 20:44:08