Stair Totals

Aim: To work out the link between the stair total and the Stair number.

In this coursework I will look at the relationship between the stair total and the position of a 3 step stair on an n x n grid.  I also must investigate further the relationship between the stair totals and the stairs on other grid size.

10 by 10 Grid

Below is an example of a 10 by 10 grid

Below is the sample of a 3 step stair extracted from the 10x10 grid

Stair total= 32+33+34+42+43+52

Stair total= 236

In order to save time in my coursework by not writing lots of words, and to help find the formula later on I will use algebra

Let T= Stair total

Let n= Stair number

So by looking at the above diagram I can say that my T value is 236 and my n value is 32.

I will begin by examining the number stairs beginning with n = 1. The reason why I am going to do this is because of the following;

• It is simple and easy to work with
• It is logical
• It will help find the nth term later because I will be starting at the position n=1

Analysis

                                                n= 1

                                                T= 1+ 2+ 3+ 11+ 12+ 21

                                                T= 50

                                                 n= 2

                                                 T= 2+ 3+ 4+ 12+ 13+ 22

                                                 T= 56

                                                 n= 3

                                                 T= 3+ 4+ 5+ 13+ 14+ 23

                                                 T= 62

                                                 n= 4

                                                 T= 4+ 5+ 6+ 14+ 15+ 24

                                                 T= 68

                                                 n= 5

                                                 T= 5+ 6+ 7+ 15+ 16+ 25

                                                 T= 74

I will now draw a table because it is at a

60

66

        + 6      + 6      + 6     + 6

Observations

From the table I have noticed a few trends which are as follows;

1. All the stair Totals(T) are even numbers
2. The stair Totals(T) increase by 6 each time

Finding the nth term

T= 6n +  

From the table I can say that When n= 5, T must be 66

                  T= 6 x 5+                  

                    66= 30+         This means that

                    66- 30=

                          36=

Testing the formula

Looking at my table when n= 2, T must = 48

I can now use my formula which I have just found to check if it is correct

                          T= 6(2) + 36

                          T= 12 + 36

                          T=48

This shows that my formula works

Proving my formula

I have noticed that on my number stairs, as you go across the number increases by 1 and as you go up the number increases by 8.  I will now illustrate this below.

           +8          

          +8

        +1         +1

I'll now do the same thing using algebra.

        + 1

T= n+ (n+ 1) + (n+ 2) + (n+ 8) + (n+ 9) + (n+ 16)

T= 6n+ 36

Relationship between stair totals

I will now find the link between the stair totals only for 3 step stairs.   I will also look for the relationship between the stair totals and the grid size.

I will now draw a table because it is at a glance data, which means that it is quick and easy to look at because the data is grouped together, and furthermore it will help me to look for patterns in the data.

         Prediction

T= 2 + 3 + 4 + 5 + 6 + 10 + 11 + 12 + 13 + 18 + 19 + 20 + 26 + 27 + 34

T= 210

Now that I have completed my coursework, I will outline the formulas that I have found during this coursework, and what it does.

Formula _                             What it does

      T= 6n + 44                                                        Formula for any 3 step stair            

           on a 10 by 10 grid

      T= 6n + 40        Formula for any 3 step stair

         on a 9 by 9 grid

      T= 6n + 36                                                         Formula for any 3 step stair

             On a 8 by 8 grid

      T= 6n + 32                                                         Formula for any 3 step stair

             On a 7 by 7 grid

      T= n                                                                   Formula for any 1 step stair

        On any grid size

      T= 3n + G + 1        Formula for any 2 step stair

        On any grid size

      T= 6n + 4G + 4        Formula for any 3 step stair

        On any grid size

      T= 10n + 10G + 10        Formula for any 4 step stair

        On any grid size

      T= 15n + 20G + 20                                            Formula for any 5 step stair

                                                                                         On any grid size

       T= 21n + 35G + 35        Formula for any 6 step stair

        On any grid size

1   x ( x + 1)          Formula for triangular  

       2        numbers

T= (Triangular number) n +         Word Formula for any

(sum of triangular numbers but 1 below stair size)G           stair size on any grid size

+ (Sum of triangular numbers but 1 below stair size)

T= ( 1x(x + 1))n +(Σ triangular numbers less 1)G           Improved word and algebraic

+ (Σ triangular numbers less 1)        formula for any stair size

       2        on any grid size

                               x-1x-1

T= ( x ( x + 1) ) n + (Σ    x (x+1) ) G +  (Σ    x (x+1) )           Algebraic formula for

      2x=12x=1any stair size on any grid

size

T= (x(x+1)) n + (Σ1(x2-1)) G + 1x(x2-1)        An improved algebraic

           2               6                  6                                     formula for any grid

                                                                                       size on any stair size