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# My aims throughout this investigation are for each step stair is to investigate the relationship between the stairs and the position of the step stair.

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Introduction

Math’s Coursework                                                                                     Step-Stairs

Math’s Coursework

The Problem

We have been set a problem which is to investigate step stairs. Throughout this report I will be planning how to tackle the problem, break it down simpler into an equation. All the results will be put into tables so I can look for a pattern. I will then produce my own rule to use and test and justify it.  The problem to which I am solving is creating my own rule to use to calculate the 3step-stair.

My aims throughout this investigation are for each step stair is to investigate the relationship between the stairs and the position of the step stair. To make sure my rule is correct I will change the step size or grid size and work out the formula again.

Aims and Objectives

• For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.
• Investigate further the relationship between the stair total and the other step stairs on other number grids.

3 Step-Stairs

Look at the stair shape drawn on the 10 by 10 number Grid below.

 91 92 93 94 95 96 97 98 99 100 841 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

This is a 3 Step Stair.

The total of the numbers inside the stair shape is:

1 + 2 + 3 + 11 + 12 + 21 = 50

The stair total for this 3 step stair is 50.

I got this by adding all the numbers within the three step stair (the bold red numbers). The letter ‘N’ represents the position of the three step stair. In this case ‘N=1’. The reason for this is because the step stair starts with the number 1. The letter ‘T’ represents the total of the step stair.

Middle

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

This is a 4 Step Stair

The total of the numbers inside the stair shape is:

1 + 2 + 3 + 4 + 11 + 12 + 13 + 21 + 22 + 31 = 120

‘T = 120’

I got this number by adding all the numbers within the 4 Step Stairs together.

The second position is 2. Like the 3 step stair you move the position one over to the right.

The total for the position 2 is:

2 + 3 + 4 + 5 + 12 + 13 + 14 + 22 + 23 + 32 = 130

‘T = 130’

Now I’ll be finding the totals if the positions were 3, 4 and 5.

 91 92 93 94 95 96 97 98 99 100 841 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   3rd 5th

PositionPosition

4th

Position

N = 3

3 + 4 + 5 + 6 + 13 + 14 + 15 + 23 + 24 + 33 = 140

T = 140

N = 4

4 + 5 + 6 + 7 + 14 + 15 + 16 + 24 + 25 + 34 = 150

T = 150

N = 5

5 + 6 + 7 + 8 + 15 + 16 + 17 + 25 + 26 + 35 = 160

T = 160

All of these positions all had one steady difference which was the number ten.

 Position (N) 1 2 3 4 5 Total (T) 120 130 140 150 160 Difference 10 10 10 10 10

This table shows the total results of each position and there differences.

The Equation will have the following:

‘T = 10n + ...’

To work out the rest of the equation I will have to minus ‘10n’ from a position.

Position 1

10n = 10

120-10 = 110

The formula for 4 step stairs is:

• T = 10n + 110

Testing

Position 2

T = 10n + 110 = 10n × 2 + 110

10 × 2 = 20

20 + 110 = 130

Position 3

T = 10n + 110 = 10n × 3 + 110

10 × 3 = 30

30 + 110 = 140

Position 4

T = 10n + 110 = 10n × 4 + 110

10 + 4 = 40

40 + 110 = 150

Position 5

T = 10n + 110 = 10n × 5 + 110

10 + 5 = 50

50 + 110 = 160

All of the totals are correct. So the formula proved successful.

5 Step Stairs

Look at the stair shape drawn on the 10 by 10 number Grid below.

 91 92 93 94 95 96 97 98 99 100 841 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

This is a 5 Step Stair

The total of the numbers inside the stair shape is:

1 + 2 + 3 + 4 + 5 + 11 + 12 + 13 + 14 + 21 + 22 + 23 + 31 + 32 + 41 = 235

'T = 235’

The second position is 2. Here is the total for it:

2 + 3 + 4 + 5 + 6 + 12 + 13 + 14 + 15 + 22 + 23 + 24 + 32 + 33 + 42 = 250

So far the difference I have spotted is 15. I am not sure if this is steady so I will investigate further so I can be sure.

 91 92 93 94 95 96 97 98 99 100 841 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   3rd 5th

PositionPosition

4th

Position

N=3

3 + 4 + 5 + 6 + 7 + 13 + 14 + 15 + 16 + 23 + 24 + 25 + 33 + 34 + 43 = 265

T=265

N=4

4 + 5 + 6 + 7 + 8 + 14 + 15 + 16 + 17 + 24 + 25 + 26 + 34 + 35 + 44 = 280

T=280

N=5

5 + 6 + 7 + 8 + 9 + 15 + 16 + 17 + 18 + 25 + 26 + 27 + 35 + 36 + 45 = 295

T=295

The difference between all these numbers is 15.

 Position (N) 1 2 3 4 5 Total (T) 235 250 265 280 295 Difference 15 15 15 15 15

Conclusion

Further Investigation

Now I have all the equations. From all the formulas I’m going to find the difference between all of them. Hopefully when I will be able to create another formula which allows me to find the total no matter what size the grid is or the position of the step stair.

Here are all the formulas for different sized step stairs.

3 step stair             T = 6n + 44 4 step stair                             T = 10n + 110 5 step stair                            T = 15n + 220 6 step stair                             T = 21n + 385 7 step stair                             T = 28n + 616 T = an + b

I have done this so that I can work out the ultimate formula.

Now I plan to find the differences between the formulas. So that if I put the formula’s together I would get the final formula.

Stem        3               4               5               6               7

(s)

44            110            220           385           616    66             110            165           231   44           55            66  11           11

From finding the difference of the numbers I did not spot the difference at first so I continued to find the difference of the results I got. As I had to continue on finding the difference I knew this wouldn’t be a quadratic equation. This is because it had three differences not two.  Now that I have the difference I realized that the formula would be cubic because it took three steps to find the difference.

The equation for this is:

• T = ½ S²

11S ³                              49 ½

6

11S ³                              -5 ½ - 11

1. 2

-5.5 – 7 1                       7 1   =   22

3        3         3 -11     -      22

2        3

3344

6         6

3 × 11       4 × 11

6              6

-11 S

6

1              2              3               4               5               6

1              3              6              10               15            21     2                 3              4                5              6    1                1                1                1

The 1st row in this diagram are all triangular numbers.

• T = ½ S ²
• ½ y ² + ½ y

3                   4                  5                     6                  7

44                110              220                 385                616     66                110                  165                231   44                 55                    66  -5.5 +11              +11 493

11 S³ 99

6            2

11   ×    3                  11 S³

6         6

1. =  11

2

This is a cubic equation because it has three differences.

The ultimate equation from previous working out is:

• T = ( 1 S² + 1 S ) N + 11 S³ - 11 S

2         2        6        6

This equation is the final equation used to calculate the total no matter what size step stair you use and the position of it.

Purn Patel

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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