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
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