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Introduction

Ben Foster        GCSE Mathematics Coursework 02/02/2008

Ben Foster

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

Centre Number --------

GCSE Coursework: Number Stairs

Teacher: -------

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.

## 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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