Just 10 Stairs!

Consider the last step Jake takes before stepping to the top step. He either steps from the $8$ th step or from the $9$ th step.

Let's let $f_n$ denote the number of ways to reach the $n$ th step by going up $1$ or $2$ steps at a time. We see that $f_{10} = f_8 + f_9$ , since there are $f_8$ ways to reach the $8$ th step and follow that with a step of $2$ , while there are $f_9$ ways to reach the $9$ th step, and follow that with a step of $1$ .

Indeed, $f_9 = f_7 + f_8$ and...oh, well, these are the Fibonacci numbers, starting with $f_1 = 1$ and $f_2 = 2$ , and $f_n = f_{n-2} + f_{n-1}$ for $n \geq 3$ .

With this indexing, $f_{10} = 89$ .

$\sum_{n=0}^{5}\binom{10-n}{n}=89$