# My Investigation is called 'Number Stairs'

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Introduction

Mathematics Coursework

## Number stairs

My Investigation is called ‘Number Stairs’

## Task Statement

I have been set the task of working out the relationship between a three-stair total and the position of the stair shape on the prearranged grid and other stair totals. For further investigation other stair totals could be worked out on different grids of our own and also different numbers of stairs other than three.

The rules for my investigation are:

1. Each stair is labelled as a number; this number is the bottom left hand number of that stair. This is known as the keystone.

2. You will work horizontally across the grid from square number one.

3. If you come to

Middle

22

23

24

25

26

27

28

15

16

17

18

19

20

21

8

9

10

11

12

13

14

1

2

3

4

5

6

7

18+19+20+25+26+32=140

This proves my formula works.

Other Grids on 3-stair totals

1 x10=6n-4

2 x10=6n+2

3 x10=6n+8

4 x10=6n+14

5 x10=6n+20

6 x10=6n+26

7x10=6n+32

8x10=6n+36

9x10=6n+40

10x10=6n+44

6 between all formulae

Prediction

12x10=6n+52

108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 |

96 | 97 | 98 | 99 | 100 | 101 | 102 | 103 | 104 | 105 | 106 | 107 |

85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |

37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |

25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 |

13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

Keystone | 1 | 2 | 3 | 4 | 5 |

Stair Total | 58 | 64 | 70 | 76 | 82 |

Nth term = dn + (a-d)

Nth term = 6n + (58-6=52)

Nth term = 6n + 52

This proves my formula is correct.

Other Stairs

4-Stair Total

10x10

Keystone | 1 | 2 | 3 | 4 | 5 |

Total | 120 | 130 | 140 | 150 | 160 |

(+10) (+10) (+10) (+10)

Nth term = dn + (a-d)

Nth term = 10n + (120-10)

Nth term = 10n + 110

Prediction:

Keystone 46

46x10= 460 460+110= 570

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 |

46+47+48+49+56+57+58+66+67+76=570

This proves that the formula works.

7x9 Square

Keystone | 1 | 2 | 3 | 4 |

Total | 90 | 100 | 110 | 120 |

(+10) (+10) (+10)

57 | 58 | 59 | 60 | 61 | 62 | 63 |

50 | 51 | 52 | 53 | 54 | 55 | 56 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

Nth term = dn + (a-d)

Nth term = 10n + (90-10=80)

Nth term = 10n + 80

Prediction:

Keystone 25

25x10=250 250+80=330

46+39+40+32+33+34+25+26+27+28=330

This proves my formula works.

8x8 Square

Keystone | 1 | 2 | 3 | 4 | 5 |

Total | 100 | 110 | 120 | 130 | 140 |

Conclusion

1

2

3

4

5

Total

175

190

205

220

235

(+15) (+15) (+15) (+15)

Nth term = dn + (a-d)

Nth term = 15n + (175-15=160)

Nth term = 15n + 160

Prediction

Keystone=24

15x24=360 360+160=520

57 | 58 | 59 | 60 | 61 | 62 | 63 |

50 | 51 | 52 | 53 | 54 | 55 | 56 |

43 | 44 | 45 | 46 | 47 | 48 | 49 |

36 | 37 | 38 | 39 | 40 | 41 | 42 |

29 | 30 | 31 | 32 | 33 | 34 | 35 |

22 | 23 | 24 | 25 | 26 | 27 | 28 |

15 | 16 | 17 | 18 | 19 | 20 | 21 |

8 | 9 | 10 | 11 | 12 | 13 | 14 |

1 | 2 | 3 | 4 | 5 | 6 | 7 |

24+25+26+27+28+31+32+33+34+38+39+40+45+46+52=520

This proves that my formula is correct.

8x8 Square

Keystone | 1 | 2 | 3 | 4 | 5 |

Total | 195 | 210 | 225 | 240 | 255 |

(+15) (+15) (+15) (+15)

Nth term = dn + (a-d)

Nth term = 15n + (195-15=180)

Nth term = 15n + 180

Prediction:

Keystone 36

36x15=540 540+180=720

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |

65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

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 |

68+60+60+52+53+54+44+45+46+47+36+37+38+39+40=720

Linking the Data of Stair Totals

8x8 5 Stair Total

Nth term = 15n + 180

7x9 5 stair Total

Nth term = 15n + 160

10x10 5 Stair total

Nth term = 15n + 220

10x10 4Stair Total

Nth term = 10n + 110

7x9 4 Stair Total

Nth term = 10n + 80

8x8 4 Stair Total

Nth term = 10n + 90

10x10 3 Stair Total

Nth term = 6n + 44

8x8 3 Stair Total

Nth term = 6n + 36

7x9 3 Stair Total

Nth term = 6n + 32

I have come to this but do not know how to link them together.

Conclusion

From my results I can see that the amount of stairs are linked with different grids.

I am not sure how it works but it does and I’ve proved it.

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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