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• Level: GCSE
• Subject: Maths
• Word count: 1945

# My Investigation is called 'Number Stairs'

Extracts from this document...

Introduction

Mathematics Coursework

## Number stairs

My Investigation is called ‘Number Stairs’

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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