Photo Credit: Ryan Farber
When observing a staircase, how many ways can you climb it? For instance, if we break it down to three stairs.. How many different ways can you climb those three steps?
One may leap two stairs at once, or take it one stair at a time. By combining those two actions, a person may step and then leap two at a time, or the reverse and leap initially and then follow it with a step. Therefore there are three ways to climb three steps: step-by-step (1+1+1), step-leap (1+2), and leap-step (2+1). However, whether we are analyzing three stairs or four stairs or five stairs, we will find that the way to climb the stairs calls for the same pattern of action: find a sum totaling n using only 1s and 2s. Since we may have any number of 1s and 2s, and the order of them in the sum matters, each solution is a composition of n with parts {1,2}.
Looking at it in purely mathematical terms, if we are trying to solve for Sn and if n is 1 then the solution is Sn=1. If n=2, there are two ways. For n=3, there are three different ways. This sequence can be generalized in the following way for n>2: the number of possibilities for n stairs is equal to the sum of Sn-1 and Sn-2, so Sn= Sn-1 + Sn-2 (the Fibonacci sequence).
http://www.maths.surrey.ac.uk/
http://books.google.com/books?
One may leap two stairs at once, or take it one stair at a time. By combining those two actions, a person may step and then leap two at a time, or the reverse and leap initially and then follow it with a step. Therefore there are three ways to climb three steps: step-by-step (1+1+1), step-leap (1+2), and leap-step (2+1). However, whether we are analyzing three stairs or four stairs or five stairs, we will find that the way to climb the stairs calls for the same pattern of action: find a sum totaling n using only 1s and 2s. Since we may have any number of 1s and 2s, and the order of them in the sum matters, each solution is a composition of n with parts {1,2}.
Looking at it in purely mathematical terms, if we are trying to solve for Sn and if n is 1 then the solution is Sn=1. If n=2, there are two ways. For n=3, there are three different ways. This sequence can be generalized in the following way for n>2: the number of possibilities for n stairs is equal to the sum of Sn-1 and Sn-2, so Sn= Sn-1 + Sn-2 (the Fibonacci sequence).
http://www.maths.surrey.ac.uk/
http://books.google.com/books?
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