Solution: 2018 Winter Midterm - 10
Author: Michiel SmidQuestion
As you all know, Nick has been dreaming to be like Spiderman. Spiderman can climb up the outside of a building;
if he is at a particular floor, then, in one step, Spiderman can move up several floors.
Since having finished marking assignment 2, Nick has been working hard to improve his Spiderman-skills: In one step, Nick can move up either two floors or three floors. (Nick lost his ability to move up one floor in one step.)
Let $n \geq 2$ be an integer and consider a building with $n$ floors, numbered $1,2,\dots,n$. (The first floor has number 1; this is not the ground floor.) Nick is standing in front of this building, at the ground level.
Let $N_n$ be the number of different ways in which Nick can climb to the $n$-th floor. Which of the following is true for any $n \geq 5$?
Since having finished marking assignment 2, Nick has been working hard to improve his Spiderman-skills: In one step, Nick can move up either two floors or three floors. (Nick lost his ability to move up one floor in one step.)
Let $n \geq 2$ be an integer and consider a building with $n$ floors, numbered $1,2,\dots,n$. (The first floor has number 1; this is not the ground floor.) Nick is standing in front of this building, at the ground level.
Let $N_n$ be the number of different ways in which Nick can climb to the $n$-th floor. Which of the following is true for any $n \geq 5$?
(a)
$N_n = N_{n-1} + N_{n-2}$
(b)
$N_n = N_{n-2} + N_{n-3}$
(c)
$N_n = N_{n-1} + N_{n-2} + N_{n-3}$
(d)
$N_n = N_{n-1} + N_{n-3}$
Solution
We can check all possibilities and sum them up.
From floor x, he goes up 2 floors: $ N_{n-2} $
From floor x, he goes up 3 floors: $ N_{n-3} $
Thus $ N_n = N_{n-2} + N_{n-3} $