Suppose Jim climbs stairs in a parking garage for exercise H

Suppose Jim climbs stairs in a parking garage for exercise. He will sometimes take two steps at a time. Let c_n be the number of ways that Jim can climb n steps.

a) Give a recurrence relation for c_n. Be sure to include the initial conditions.

b) Use this recurrence relation to calculate in how many ways Jim can climb a flight of 12 steps.

This needs to be related to tecurrence relations for example c_{n =} c_n

Solution

(a)

If Jim attained (n-1)th stair then he has only one way to attain the nth stair, by stepping one up. So he has here
1*C(n-1) ways

If Jim attained (n-2)th stair then he has two ways to get to the nth stair, one big step or two small ones. But the two small ones first leads to the (n-1)th stair which we counted already. So we need only to count the one big step case. Thus there are 1*C(n-2) ways

thus the function could be written as :

Cn = C(n-1) + C(n-2).

(b)

Jim can only attain step 0 in c(0)=1 way

and step 1 in c(1)=1 way

therefore

C(12) = 233

therefore total number of ways JIm could . climb are = 233 ways