Question

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one, two, or three stairs at a time. b) What are the initial conditions? c) In many ways can this person climb a flight of eight stairs?

Solution

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(a) Let $a_n$ represent the number of ways to climb $n$ stairs if the person climbing the stairs can take one, two, or three stairs at a time.

$\textbf{First case}$ The final step is taken by 1 stair, then there are $a_{n-1}$ ways to climb the first $n-1$ steps.

$\textbf{Second case}$ The final step is taken by 2 stairs, then there are $a_{n-2}$ ways to climb the first $n-2$ steps.

$\textbf{Third case}$ The final step is taken by 3 stairs, then there are $a_{n-3}$ ways to climb the first $n-3$ steps.

Adding the number of sequences of all three cases, we then obtain:

$a_n=a_{n-1}+a_{n-2}+a_{n-3}$

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