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• Level: GCSE
• Subject: Maths
• Word count: 5404

# Number Stairs

Extracts from this document...

Introduction

Number coursework 1

In this piece of coursework I am going to investigate the relationship between the stair total and the position of the stair shape on the grid. I am also going to further investigate about the stair total and grid size.

I will use (T) for my stair total, and (N) for my stair number.

 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 21 11 12 1 2 3

Stair number (N), which is 1.

By calculating the sum of all figures inside the stairs gives you the stair total.The symbol for the stair total is (T). With the stair number 1 we get a stair total of 1+2+3+11+12+21=50. The stair total is calculated accordingly to the stair number.

 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10
 22 12 13 2 3 4

Stair number (N), which is 2.

Thus our stair total for this stair grid in the 10x10 grid is, 2 + 3 +4 +12 +13 +22 = 56. Therefore T = 56.

91

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81

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84

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71

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80

Middle

Here is an 8x8 grid showing the stair total and the stair number:

 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
 17 9 10 1 2 3

Stair number (n)=1

By calculating the sum of all figures inside the stairs gives you the stair total. With the stair number 1 we get a stair total of 1+2+3+9+10+17= 42 =T. The stair total is calculated accordingly to the stair number for any grid size. This is for the 8x8 grid with the stair number 1.

 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
 18 10 11 2 3 4

Stair number= 2

Whereas the stair total= 2+3+4+10+11+18 = 48 = T

 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
 19 11 12 3 4 5

Stair number=3

Whereas the stair total= 3+4+5+11+12+19= 54 = T

The following table shows the stair total (T) depending on the relevant stair number for the 8x8 grid. Which are from 1 to 5.

 N T 1 42 2 48 3 54 4 60 5 66

42    48    54   60    66

+6                                  +6                                  +6                              +6

It is clear that the difference between the numbers is 6. This means that the formula, to solve the total, must include a multiple of 6. This will lead to the fact that the nth term has to have 6n in the formula, the extra is calculated by working out what is left over this will be +36.Therefore, the formula has to be T =6n + 36. By substituting the stair number to the nth term we get the stair total. Here we can see that is clearly evident that the remaining part in the nth term for the 8x8 grid has decreased by 4 as compared to the 9x9 grid.  The way how this formula works is the following:

 N 6n + 36 T 1 (6x1) + 36 42 2 (6x2) + 36 58 3 (6x3) + 36 54 4 (6x4) + 36 60 5 (6x5) + 36 66

Which means that (6 x STAIR NUMBER) + 40 = STAIR TOTAL

Now that I have worked out the formula for the 8x8 grid I am going to use the formula in random staircases with random stair numbers. And those are:

 62 54 55 46 47 48

Here we can see that stair number= 46

Whereas stair total= 46+47+48+54+55+62=312

But by using the nth term, stair total= (6x46) + 36= 312

 57 49 50 41 42 43

Stair number=41

Whereas stair total= 41+42+43+49+50+57= 282

But by using the nth term, stair total= (6x41) + 36= 282

 43 35 36 27 28 29

Stair number=27

Whereas stair total= 27+28+29+35+36+43=198

Conclusion

9+9+1+9+10= 3(9) +11= 38

Here is another example:

 89 79 80
 n+10 n n+1

Total for algebraic staircase= n+n+1+n+10 = 3n + 11

We can also evaluate that Stair number (n) = 79

By substitution stair total= 79+79+1+79+10= 3(79) +11= 248

Now I am going to begin my investigation with the 9x9 grid size for the 2 step stair case:

 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9
 10 1 2

Here we can see that the stair number = 1

Therefore the stair total = 1+2+10=13

 11 2 3

Stair number = 2, therefore, stair total = 2+3+11= 16

 12 3 4

Stair number = 3, therefore stair total = 3+4+12= 19

The following table shows the stair total (T) depending on the relevant stair number for the 7x7 grid, with stair case number4. Which are from 1 to 5.

 N T 1 13 2 16 3 19 4 22 5 25

13        16      19        22      25

+3                                  +3                               +3                           +3

Now that I solve the total, must include a multiple of 10. This will lead to the fact that the nth term has to have 3n in the formula, the extra is calculated by working out what is left over this will be +10.Therefore, the formula has to be T =3n + 10. By substituting the stair number to the nth term we get the stair total. The way how the formula was in algebraic terms is:

Formula to find out and nth term of any linear sequence = dn + (a-d) =T

a= First term in the linear sequence = 13

d= common difference = 3

n=stair number

T = stair total

By substitution, nth term was formed by: (3)(n) + (13-3) = 10n + 11

The way how the formula works is the following:

 N 3n + 10 T 1 (3x1) + 10 13 2 (3x2) + 10 16 3 (3x3) + 10 29 4 (3x4) + 10 22 5 (3x5) + 10 25

Which means that (3 x STAIR NUMBER) + 10 = STAIR TOTAL

 n+9 n n+1

Here is an alternative way to find the stair total of the 9x9 grid by using further algebraic method for the 2 step stair case.

 17 8 9

As we can see here n=stair number, and the 2x2 stair case from the 9x9 grid can be substituted in to the formula staircase for the 9x9 grid.

Total for algebraic staircase= n+n+1+n+9 = 3n + 10

We can also evaluate that Stair number (n) = 8

 n+9 n n+1

By substitution stair total= 8+8+1+8+9= 3(8) +10= 34

Here is another example:

 80 71 72

Total for algebraic staircase= n+n+1+n+9 = 3n + 10

We can also evaluate that Stair number (n) = 71

By substitution stair total= 71+71+1+71+9= 3(71) +10= 223

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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