I'm trying to generalize the problem that follows oif the number of stairs is "n", the problem is "You climb either 1 or 2 stairs at a time, at any given time. How many ways can you get up 11 stairs?"
Eventually I noticed that as I tried n = 1,2,3,4,5 the number of possible ways are 1, 2, 3, 5, 8.... which is somehow resembling the fibonacci sequence... I tried to prove using pascal triangle since it is comnected to combinatorics and fibonacci but i dunno how to start... any idea?
To climb $n$ stairs you can either climb $n-1$ stairs and step up one or climb $n-2$ stairs and step up two. If $A(n)$ is the number of ways to climb $n$ stairs, this says $A(n)=A(n-1)+A(n-2)$, which is exactly the Fibonacci recurrence.