# Mathematics - Number Stairs

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Introduction

Ben Foster GCSE Mathematics Coursework 02/02/2008

Ben Foster

-------- School---------

Centre Number --------

GCSE Coursework: Number Stairs

Teacher: -------

Table of Contents

Aim

First Part

3 Step-Staircase / Grids of width 10

Second Part

3 Step-Staircase

3 Step-Staircase / Grid Width 8

3 Step-Staircase / Grid Width 9

3 Step-Staircase / Grid Width 11

3 Step-Staircase / Grid Width 12

2 Step-Staircase

2-Step Staircase/ Grid Width 8

2-Step Staircase/ Grid Width 9

2-Step Staircase/ Grid Width 10

2-Step Staircase/ Grid Width 11

2-Step Staircase/ Grid Width 12

1 Step-Staircase

4 Step-Staircase

4 Step-Staircase / Grid Width 8

4 Step-Staircase / Grid Width 9

4 Step-Staircase / Grid Width 10

4 Step-Staircase / Grid Width 11

4 Step-Staircase / Grid Width 12

5 Step-Staircase

5 Step-Staircase / Grid Width 8

5 Step-Staircase / Grid Width 9

5 Step-Staircase / Grid Width 10

5 Step-Staircase / Grid Width 11

5 Step-Staircase / Grid Width 12

The Final Formula

Finding the “n” term of the formula

Finding the “g” term of the formula

Finding the number term of the formula

## Aim

In this investigation, I will be aiming to find out the formulas of the combinations of the size of number stairs and the size of grid widths. Eventually I will convene a formula connecting the stair totals and other steps on other number grids.

## First Part

## 3 Step-Staircase / Grids of width 10

21 | ||

11 | 12 | |

1 | 2 | 3 |

n | 1 | 2 | 3 | 4 | 5 |

T | 50 | 56 | 62 | 68 | 74 |

There is a pattern here, so a formula may be constructed:

It is going up by 6 every time thus making it “6n”

If the 1st term is 50 then the 0th term is 44

So therefore the formula here is T = 6n+44

To make sure this formula works, the n term can be substituted with a number, for example, 20

T = 6 x 31 + 44 = 230

So:

51 | ||

41 | 42 | |

31 | 32 | 33 |

31 + 32 + 33 + 42 + 41 + 51 = 230

So therefore my prediction works but I must prove it algebraically:

n+20 | ||

n+10 | n+11 | |

n | n+1 | n+2 |

n + (n+1) + (n+2)

Middle

12

1

T = n

T = n

T = n

T = n

T = n

2

T = 3n + 9

T = 3n + 10

T = 3n + 11

T = 3n + 12

T = 3n + 13

3

T = 6n + 36

T = 6n + 40

T = 6n + 44

T = 6n + 48

T = 6n + 52

4

5

## 4 Step-Staircase

### 4 Step-Staircase / Grid Width 8

25 | |||

17 | 18 | ||

9 | 10 | 11 | |

1 | 2 | 3 | 4 |

n | 1 | 2 | 3 | 4 | 5 |

T | 100 | 110 | 120 | 130 | 140 |

Suspected formula: T = 10n + 90

Prediction / Test: 10 x 25 + 90 = 340

49 | |||

41 | 42 | ||

33 | 34 | 35 | |

25 | 26 | 27 | 28 |

25 + 26 + 27 + 28 + 33 + 34 + 35 + 41 + 42 + 49 = 340

Algebraic Proof:

n+24 | |||

n+16 | n+17 | ||

n+8 | n+9 | n+10 | |

n | n+1 | n+2 | n+3 |

n + (n+1) + (n+2) + (n+3) + (n+8) + (n+9) + (n+10) + (n+16) + (n+17) + (n+24) = 10n + 90

8 | 9 | 10 | 11 | 12 | |

1 | T = n | T = n | T = n | T = n | T = n |

2 | T = 3n + 9 | T = 3n + 10 | T = 3n + 11 | T = 3n + 12 | T = 3n + 13 |

3 | T = 6n + 36 | T = 6n + 40 | T = 6n + 44 | T = 6n + 48 | T = 6n + 52 |

4 | T = 10n + 90 | ||||

5 |

### 4 Step-Staircase / Grid Width 9

28 | |||

19 | 20 | ||

10 | 11 | 12 | |

1 | 2 | 3 | 4 |

n | 1 | 2 | 3 | 4 | 5 |

T | 110 | 120 | 130 | 140 | 150 |

Suspected formula: T = 10n + 100

Prediction / Test: 10x 24 + 100 = 340

51 | |||

42 | 43 | ||

33 | 34 | 35 | |

24 | 25 | 26 | 27 |

24 + 25 + 26 + 27 + 33 + 34 + 35 + 42 + 43 + 51 = 340

Algebraic Proof:

n+27 | |||

n+18 | n+19 | ||

n+9 | n+10 | n+11 | |

n | n+1 | n+2 | n+3 |

n + (n+1) + (n+2) + (n+3) + (n+9) + (n+10) + (n+11) + (n+18) + (n+19) + (n+27) = 10n + 100

8 | 9 | 10 | 11 | 12 | |

1 | T = n | T = n | T = n | T = n | T = n |

2 | T = 3n + 9 | T = 3n + 10 | T = 3n + 11 | T = 3n + 12 | T = 3n + 13 |

3 | T = 6n + 36 | T = 6n + 40 | T = 6n + 44 | T = 6n + 48 | T = 6n + 52 |

4 | T = 10n + 90 | T = 10n + 100 | |||

5 |

### 4 Step-Staircase / Grid Width 10

31 | |||

21 | 22 | ||

11 | 12 | 13 | |

1 | 2 | 3 | 4 |

n | 1 | 2 | 3 | 4 | 5 |

T | 120 | 130 | 140 | 150 | 160 |

Suspected formula: T = 10n + 110

Prediction / Test: 10 x 23 + 110 = 340

53 | |||

43 | 44 | ||

33 | 34 | 35 | |

23 | 24 | 25 | 26 |

23 + 24 + 25 + 26 + 33 + 34 + 35 + 43 + 44 + 53 = 340

Algebraic Proof:

n+30 | |||

n+20 | n+21 | ||

n+10 | n+11 | n+12 | |

n | n+1 | n+2 | n+3 |

n + (n+1) + (n+2) + (n+3) + (n+10) + (n+11) + (n+12) + (n+20) + (n+21) + (n+30) = 10n + 110

8 | 9 | 10 | 11 | 12 | |

1 | T = n | T = n | T = n | T = n | T = n |

2 | T = 3n + 9 | T = 3n + 10 | T = 3n + 11 | T = 3n + 12 | T = 3n + 13 |

3 | T = 6n + 36 | T = 6n + 40 | T = 6n + 44 | T = 6n + 48 | T = 6n + 52 |

4 | T = 10n + 90 | T = 10n + 100 | T = 10n + 110 | ||

5 |

From here, I see a pattern of a jump size of 10 so I predict for grid widths 11 and 12 will be T = 10n + 120 and T = 10n + 130.

### 4 Step-Staircase / Grid Width 11

34 | |||

23 | 24 | ||

12 | 13 | 14 | |

1 | 2 | 3 | 4 |

n | 1 | 2 | 3 | 4 | 5 |

T | 130 | 140 | 150 | 160 | 170 |

Suspected formula: T = 10n + 120

Prediction / Test: 10 x 47 + 120 = 590

80 | |||

69 | 70 | ||

58 | 59 | 60 | |

47 | 48 | 49 | 50 |

47 + 48 + 49 + 50 + 58 + 59 + 60 + 69 + 70 + 80 = 590

Algebraic Proof:

n+33 | |||

n+22 | n+23 | ||

n+11 | n+12 | n+13 | |

n | n+1 | n+2 | n+3 |

n + (n+1) + (n+2) + (n+3) + (n+11) + (n+12) + (n+13) + (n+22) + (n+23) + (n+33) = 10n + 120

8 | 9 | 10 | 11 | 12 | |

1 | T = n | T = n | T = n | T = n | T = n |

2 | T = 3n + 9 | T = 3n + 10 | T = 3n + 11 | T = 3n + 12 | T = 3n + 13 |

3 | T = 6n + 36 | T = 6n + 40 | T = 6n + 44 | T = 6n + 48 | T = 6n + 52 |

4 | T = 10n + 90 | T = 10n + 100 | T = 10n + 110 | T = 10n + 120 | |

5 |

### 4 Step-Staircase / Grid Width 12

37 | |||

25 | 26 | ||

13 | 14 | 15 | |

1 | 2 | 3 | 4 |

n | 1 | 2 | 3 | 4 | 5 |

T | 140 | 150 | 160 | 170 | 180 |

Suspected formula: T = 10n + 130

Prediction / Test: 10 x 15 + 130 = 280

51 | |||

39 | 40 | ||

27 | 28 | 29 | |

15 | 16 | 17 | 18 |

15 + 16 + 17 + 18 + 27 + 28 + 29 + 39 + 40 + 51 = 280

Algebraic Proof:

n+36 | |||

n+24 | n+25 | ||

n+12 | n+13 | n+14 | |

n | n+1 | n+2 | n+3 |

n + (n+1) + (n+2) + (n+3) + (n+12) + (n+13) + (n+14) + (n+24) + (n+25) + (n+36) = 10n + 130

8 | 9 | 10 | 11 | 12 | |

1 | T = n | T = n | T = n | T = n | T = n |

2 | T = 3n + 9 | T = 3n + 10 | T = 3n + 11 | T = 3n + 12 | T = 3n + 13 |

3 | T = 6n + 36 | T = 6n + 40 | T = 6n + 44 | T = 6n + 48 | T = 6n + 52 |

4 | T = 10n + 90 | T = 10n + 100 | T = 10n + 110 | T = 10n + 120 | T = 10n + 130 |

5 |

## 5 Step-Staircase

### 5 Step-Staircase / Grid Width 8

33 | ||||

25 | 26 | |||

17 | 18 | 19 | ||

9 | 10 | 11 | 12 | |

1 | 2 | 3 | 4 | 5 |

n | 1 | 2 | 3 | 4 | 5 |

T | 195 | 210 | 225 | 240 | 255 |

Suspected formula: T = 15n + 180

Prediction / Test: 15 x 34 + 180 = 690

66 | ||||

58 | 59 | |||

50 | 51 | 52 | ||

42 | 43 | 44 | 45 | |

34 | 35 | 36 | 37 | 38 |

34 + 35 + 36 + 37 + 38 + 42 + 43 + 44 + 45 + 50 + 51 + 52 + 58 + 59 + 66 = 690

Algebraic Proof:

n+32 | ||||

n+24 | n+25 | |||

n+16 | n+17 | n+18 | ||

n+8 | n+9 | n+10 | n+11 | |

n | n+1 | n+2 | n+3 | n+4 |

n + (n+1) + (n+2) + (n+3) + (n+4) + (n+8) + (n+9) + (n+10) + (n+11) + (n+16) + (n+17) + (n+18) + (n+24) + (n+25) + (n+32) = 15n + 180

8 | 9 | 10 | 11 | 12 | |

1 | T = n | T = n | T = n | T = n | T = n |

2 | T = 3n + 9 | T = 3n + 10 | T = 3n + 11 | T = 3n + 12 | T = 3n + 13 |

3 | T = 6n + 36 | T = 6n + 40 | T = 6n + 44 | T = 6n + 48 | T = 6n + 52 |

4 | T = 10n + 90 | T = 10n + 100 | T = 10n + 110 | T = 10n + 120 | T = 10n + 130 |

5 | T = 15n + 180 |

### 5 Step-Staircase / Grid Width 9

37 | ||||

28 | 29 | |||

19 | 20 | 21 | ||

10 | 11 | 12 | 13 | |

1 | 2 | 3 | 4 | 5 |

n | 1 | 2 | 3 | 4 | 5 |

T | 215 | 230 | 245 | 260 | 275 |

Suspected formula: T = 15n + 200

Prediction / Test: 15 x 39 + 200 = 785

75 | ||||

66 | 67 | |||

57 | 58 | 59 | ||

48 | 49 | 50 | 51 | |

39 | 40 | 41 | 42 | 43 |

39 + 40 + 41 + 42 + 43 + 48 + 49 + 50 + 51 + 57 + 58 + 59 + 66 + 67 + 75 = 785

Algebraic Proof:

n+36 | ||||

n+27 | n+28 | |||

n+18 | n+19 | n+20 | ||

n+9 | n+10 | n+11 | n+12 | |

n | n+1 | n+2 | n+3 | n+4 |

Conclusion

Staircase Size | 1 | 2 | 3 | 4 |

Formula | 0g+0 | g+1 | 4g+4 | 10g+10 |

Row 1 | g+1 3g+3 6g+6 | |||

Row 2 | 2 3 | |||

Row 3 | 1 |

After finding out the method of differences, it turns out to be exactly the same sequence for finding out the “g” term considering the differences between the staircase sizes of the number terms follow the same sequence as the “g” term!

So now, there is a constant difference of 1, therefore, 1 ÷ 6 = 1/6s³

Now, I would expect a quadratic but since the sequence is exactly the same as the “g” term formula, I am predicting an automatic linear sequence:

Staircase Size | 1 | 2 | 3 | 4 |

1/6³ Formula | 0-(1/6 x 1³) | 1-(1/6 x 2³) | 4-(1/6 x 3³) | 10-(1/6 x 4³) |

Formula Solution | [-1/6] | [-1/3] | [-1/2] | [-2/3] |

Row 1 | [-1/6] [-1/6] [-1/6] |

As predicted, the second part to the formula is -1/6.

The overall product is now: TS = n(0.5s² + 0.5s) + g(1/6s³ - 1/6s) + (1/6s³ - 1/6s)

To neaten the formula up a bit:

This seems ludicrous at first so I will now substitute the letters with numbers to prove this formula works.

With “n” being 1, “s” being 3 and “g” being 10, I should come up with a total step number of 50.

Staircase Size = 3, Grid width = 10, bottom left number = 1

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

So I have seen that this works fully and now that I have the overall formula connecting the staircase sizes, the grid widths and the bottom left number of the staircases together; the total of the numbers in any staircase of any size of any grid width can be worked out by using this formula:

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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