We just finished induction and are now starting on Recursive functions. The assigned problem is: How many ways is it possible to climb a staircase if n steps if one is allowed to take either one or two steps at a time?
Call $\displaystyle a_n$ the ways of getting to the nth step.
Suppose we want to climb to the nth step, and $\displaystyle n\geq{2}$.
There are 2 possible ways of getting there:
- We are at the n-1 step, and we jump to the next. $\displaystyle a_{n-1}$ ways of doing this, since we have to get to the n-1 step
- We are at the n-2 step, and we jump directly to n. $\displaystyle a_{n-2}$ ways of doing this
So we have $\displaystyle a_{n}=a_{n-1}+a_{n-2}$
Now $\displaystyle a_0=1$ (there's one way of doing nothing) and $\displaystyle a_1=1$ and our sequence is determined