Question

Set 5 a. A staircase has 10 steps. You walk up taking one or two at a time. How many ways can you go up? We have n dollars to spend. Every day we either buy a candy for 1 dollar, or an ice cream for 2 dollars. In how many ways can we spend the money? Explain for n-5, and then conjecture for n dollars. Prove your conjecture. Define Fibonacci numbers completely. Why do you need two initial values? In the problem (a) above, how many ways can you go up if you walk up taking one, two or three at a time? Prove that if n is a multiple of 4, then F, is a multiple of 3. b. c. d. e.

Answer 1

d) Let’s let N_k denote the number of ways to climb k stairs in the manner described. (So we’re looking for N₁₀.) Notice that for k ≥ 4 there are 3 “moves” one can do for your first step: you can climb 1,2, or 3 stairs. If you climb 1 stair then there are N_k−1 ways to finish; if you climb 2 stairs there are Nk−2 ways to finish; and if you climb 3 stairs there are N_k−3 ways to finish. Thus:

N_k = N_k−1 + N_k−2 + N_k−3

Well, it’s pretty easy to see that for the k < 4 we have N₁ = 1, N₂ = 2 and N₃ = 4, so using the above we can calculate N₁₀ using

So there are 274 ways to climb the stairs.

c)  F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34....................

The initial values F₀=0 and F₁=1 are part of the definition of the Fibonacci sequence. They can't be derived, because e.g. you could just as well pick any two numbers and apply the same recursion relation to get a different sequence. They're just the simplest numbers to start with.

b) Candy = 1

ice cream = 2

Let take n = 1 ==> 1candy = 1way

Let take n = 2 ==> 2candy or 1 icecream = 2 ways

Let take n = 3 ==> 3candy or 1candy+1icecreamz = 2ways

Let take n = 4 ==> 4 candy or 2candy+1icecream or 2icecream = 3ways

Let take n = 5 ==> 5C or 3C + 1I or 1C + 2I = 3ways

Let take n = 6 ==> 6C or 4C + 1I or 2C + 2I or 3I = 4ways

Let take n = 7 ==> 7C or 5C + 1I or 3C + 2I or C+3I = 4ways

Let take n = 8 ==> 8C or 6C + 1I or 4C + 2I or 2C + 3I +  4I = 5ways

From the above we can say that the formula is

ways

a) Let Nk be the number of ways to get up k stairs, where you can either take one or two steps at a time. Obviously N(1)=1 because when you have one stair left all you can do is walk up it.

With two stairs left to go, we can go up both stairs at once or we can go up one stair and be on stair 1. So N(2)=1+N(1)=1+1=2

Similarly as above problem

So there are 89 ways to climb the stairs.

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