2005 Alabama ARML TST Problems/Problem 10
Problem
When Jon Stewart walks up stairs he takes one or two steps at a time. His stepping sequence is not necessarily regular. He might step up one step, then two, then two again, then one, then one, and then two in order to climb up a total of 9 steps. In how many ways can Jon walk up a 14 step stairwell?
Solution
Solution 1
He can either take 14 one-steps, or 12 one-steps and 1 two-step, etc., so we have
Solution 2
Let represent the number of sequences of steps that can be taken up the stairs. Clearly . Any sequence of length is either composed of a sequence of length followed by another step or a sequence of length followed by two steps. Thus we have the recursion which indeed is the Fibonacci sequence, except with indexes shifted. Matching them, we have , and .
See also
2005 Alabama ARML TST (Problems) | ||
Preceded by: Problem 9 |
Followed by: Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |