# Random Sample.

Extracts from this document...

Introduction

Ahmed Alaskary

GSCE COURSEWORK

## PART 1

Random Sample

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 |

I drew 5 3-step stairs all of them were the same size, but they differed in position.

Number Of Shape | Calculations | Total |

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

8 | 8+9+10+18+19+28 | 92 |

34 | 34+35+36+44+45+54 | 248 |

71 | 71+72+73+81+82+91 | 470 |

78 | 78+79+80+88+89+98 | 512 |

Comment:

1) Notice how all the totals are even

2) The totals increase the higher up the grid you go

3) Across goes up about 42 (92-50)

4) Up goes up about 420 (470-50)

Systematic Sample

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 |

I drew 5 stairs all on the bottom line going across, I did this to see if there was a pattern.

Number Of Shape | Calculations | Total |

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

2 | 2+3+4+12+13+22 | 56 |

3 | 3+4+5+13+14+23 | 62 |

4 | 4+5+6+14+15+24 | 68 |

5 | 5+6+7+15+16+25 | 74 |

Comment:

There is clearly a pattern here; the totals are increasing by 6, we call this a linear sequence.

From this information I predict that the next total will be 80 and the 8th will be 92.

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 |

Results:

Number Of Shape | Calculations | Total |

6 | 6+7+8+16+17+26 | 80 |

8 | 8+9+10+18+19+28 | 92 |

Both my predictions were correct.

Algebraic Expression

X+20 | ||

X+10 | X+11 | |

X | X+1 | X+2 |

I drew 1 stair with Algebra on it.

Total:

(x)+(x+1) +(x+2) +(x+10) +(x+11) +(x+20) = 6X+44

Comment:

The reason it’s 6x is because there are 6 stairs (x+x+x+x+x+x)

It’s 44 because the totals across for these stairs are 4 acrossand 40 upwards.

4: 1+1+2=4

1: This 1 came from the 11, because you go 1 across from the 10 to get 11

40: 20+10+10=40

10: This 10 came from the 11, because you go 10 up from the 1 to get 11

## PART 2

21 | 22 | 23 | 24 | 25 |

16 | 17 | 18 | 19 | 20 |

11 | 12 | 13 | 14 | 15 |

6 | 7 | 8 | 9 | 10 |

1 | 2 | 3 | 4 | 5 |

31 | 32 | 33 | 34 | 35 | 36 |

25 | 26 | 27 | 28 | 29 | 30 |

19 | 20 | 21 | 22 | 23 | 24 |

13 | 14 | 15 | 16 | 17 | 18 |

7 | 8 | 9 | 10 | 11 | 12 |

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

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 |

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 |

Middle

56

57

58

59

60

61

62

63

46

47

48

49

50

51

52

53

54

37

38

39

40

41

42

43

44

45

28

29

30

31

32

33

34

35

36

19

20

21

22

23

24

25

26

27

10

11

12

13

14

15

16

17

18

1

2

3

4

5

6

7

8

9

Results:

Name Of Grid | Calculations | Total |

9 | 1+2+3+10+11+19 | 46 |

My prediction was correct.

Algebraic Expression

x+g+g | ||

x+g | x+g+1 | |

x | x+1 | x+1+1 |

I have drawn a 3-step stair with algebra on it, the g represents grid.

Totals:

(x)+(x+1) +(x+1+1) +(x+g) +(x+g+1) +(x+g+g) = 6x+4g+4

Comment:

The reason it’s 6x is because there are 6 stairs, each one with one x on it (x+x+x+x+x+x=6x), x represents the name of the shape.

The reason it’s 4g is because say for example you had a 5 grid, notice how every time you go up the total increases in accordance with the grid number; e.g.:

21 | 22 | 23 | 24 | 25 |

16 | 17 | 18 | 19 | 20 |

11 | 12 | 13 | 14 | 15 |

6 | 7 | 8 | 9 | 10 |

1 | 2 | 3 | 4 | 5 |

+5

So basically if it were any grid the number on top of one would always increase by the grid number. That is why the g is situated on top of the x in my algebra stairs; and since there are 4g’s we write it down in the formula as 4g.

The reason it’s +4 is simply because it’s (1+1+1+1). These ‘1’s’ are on my algebra stairs because in any stairs no matter what grid it is, you always go one across.

21 | 22 | 23 | 24 | 25 |

16 | 17 | 18 | 19 | 20 |

11 | 12 | 13 | 14 | 15 |

6 | 7 | 8 | 9 | 10 |

1 | 2 | 3 | 4 | 5 |

+1

So, my formula is: 6x+4g+4

I will now experiment it on 3 different grids to see if it works.

I will try it on a 3 grid, a 7 grid and a 10 grid.

7 | 8 | 9 |

4 | 5 | 6 |

1 | 2 | 3 |

Name Of Grid | Calculations | Total |

3 | 1+2+3+4+5+7 | 22 |

Name Of Grid | Key | Calculations | Total |

3 | x=1 g=3 | (6x1)+(4x3)+4 | 22 |

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 |

Name Of Grid | Calculations | Total |

7 | 5+6+7+12+13+19 | 62 |

Name Of Grid | Key | Calculations | Total |

7 | x=5 g=7 | (6x5)+(4x7)+4 | 62 |

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 |

Name Of Grid | Calculations | Total |

10 | 45+46+47+55+56+65 | 314 |

Name Of Grid | Key | Calculations | Total |

10 | x=45 g=10 | (6x45)+(4x10)+4 | 314 |

My formula worked for all 3 grids, from this I can conclude that the formula, 6x+4g+4 will work on any grid provided that it’s a 3 step-stair.

The formula 6x+4g+4 is a linear formula. I have found a formula that will work on any 3-step stair, now I need to find a formula which will work on any sized step.

I will now change the stairs size to try and find a formula which will work on any stair size.

I will do a 3, 4, 5 and 6 step stairs on a 10 grid.

3-STEP STAIR

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 |

4-STEP STAIR

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 |

Conclusion

10 x 3= 3 x 10 Notice how these are triangle numbers. However, it’s the value of n+1.

20 x 3= 4 x 15 So we substitute it with the triangle numbers formula except we use (n+1):

35 x 3= 5 x 21 [n (n+1) (n+1) +1)]/2, but remember we have to divide by 3 because we

multiplied by 3 earlier so the formula is: [n (n+1) (n+2)]/6

In our series the 4 is from a 3 step, the nth step would need to be (n-1) in

the above formula (Blue). We substitute n with (n-1):

[(n-1) ((n-1) +1) ((n-1) +2)]/6

We can simplify the formula into this:

[n (n+1) (n-1)]/6

So the universal formula is: [n (n+1)]/2 x + [n (n+1) (n-1)]/6 g+ [n (n+1) (n-1)]/6

n= step size

x= number of shape

g= grid number

I will now test to see if it works, I want to get the formula for a 3-step stair, so I do the following:

[3(3+1)]/2 x + [3(3+1)(3-1)]/6 g + [3(3+1) (3-1)]/6 which equals:

6x+4g+4 – My formula works for the 3-step stairs.

I will do the same, but this time for a 7 step stairs:

[7(7+1)]/2 x + [7(7+1) (7-1)]/6 g + [7(7+1) (7-1)]/6 which equals:

28x+56g+56

Clearly my formula works, through the use of algebra and my insight into the relationships of series and triangle numbers I was able to achieve this. I needed to use a variety of methods and techniques to complete the universal formula, always linking it back with previous formulas and tables.

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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