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number stairs

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Introduction

Nazma Khan 10A

Maths coursework image00.png

Number Stairs

In this coursework, I will be investigating the relationship between the stair totals and the position of the stair shape on the grid. I will be using the 3 step stair on a 10 by 10 grid to find out a relationship, which will help me to solve any problem regarding to solve the 3 step stair. The 10 by 10 grid is order in the reverse order from 1-100, starting from 1 situated at the bottom left hand corner and the 100 at the top right hand corner. However in order to see a sequence I would be carrying out this investigation in a systematic order in order to see a pattern. The way in which I will be systematic is by starting from the smallest number and the moving 1 box to the right each time.

This is how the grid looks like on a 10 by 10 grid and also how the 3 step stair looks like:

91

92

93

94

95

96

97

98

99

100image01.png

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

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Middle

image02.png

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        17+18+19+27+28+37= 146

As you can see I have moved the step stair systematically 1 step towards the right, and also I first started off with the smallest number first, in order to see a pattern emerging.

Therefore the difference is 6. So you multiply 6 x 14 =84

Then to get the answer 128 you + 44.              84 + 44= 128

Furthermore I will be extending my investigation further by changing the size of steps, e.g. 1 step stair, 2 step stair…, on a 10 by 10 grid. I believe that as a result to this it will allow me to see a link between the stair total, therefore I will be trying to get an overall formula which will work for the entire step stair on a 10 by 10 grid.

I will be using the same method but in algebra form.

image04.png

        χ        

image04.png

image04.pngimage04.png

        χ+χ+1+χ+10 = 3χ + 11

image04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.png

                             χ+χ+1+χ+2+χ+10+χ+11+χ+20= 6χ+44

image04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.png

                 χ+χ+1+χ+2+χ+3+χ+10+χ+11+χ+12+χ+20+χ+21+χ+30 =10χ+110

image04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.pngimage04.png

        15χ+1+2+3+4+10+11+12+13+20+21+22+30+31+40=

                                                        15χ+220

Now that I have found the formula for the step stairs shown above, I will be taking it further in order to get an overall formula for any size step stair which works on a 10 by 10 grid.

...read more.

Conclusion

(½ n² + ½ n) χ+ 11/3 (n-1) (1/2 n²+1/2 n)

(1/2 16 + ½ 4) χ + 11/3 (4-1) (1/2 16 + ½ 4)

10χ + 11/3 x 3 x 10

=10 χ + 110

This was to find out the 4 step stair.

Now to check if it works for 8 step stair

(1/2 8 + 4) χ+ 11/3 (8-1) (½ 8 + ½ 8)

(32 + 4) χ + 11/3 (7) (32+44)

     36X 11/3 χ 7 X 36

     36X + 924                           Therefore I have found out a formula for 8 step stair

Now I am going to find out a formula for any size step on any size grid.

image04.png

image04.pngimage04.png

        χ +χ+1+χ+2+χ+3+χ+A+χ+A+1+χ+A+2+χ+2A+χ+2A+image04.pngimage04.pngimage04.png

                                        χ+3A.image04.pngimage04.pngimage04.pngimage04.png

Step stair                               Formula                Difference

n=1                                        χ       +0                    

                                          1              

n=2                                        3χ+A+1

                                                                                    3                    

n=3                                        6χ+4A+4

                                                                                    6

n=4                                        10χ+10A+10

                                                                                    10

n=5                                        15χ+20A+20

                                                                                    15

n=6                                        21χ+35A+35

Therefore by looking at the differences I can see that it is the same as the co=efficient of χ. Therefore the formula for any size grid and step stair is:

T= (1/2 n² + ½ n) χ+ 1/3 (n-1) (1/2 n² + 1/2n) a + 1/3 (n-1) (1/n² + 1/2n)

This is the formula for the grid. However I believe that you can simplify this formula which now looks like this:

T= (1/2 n² + ½ n) χ+ 1/3 (n-1) a + 1/3 (n-1)

You can simplify this formula even further:

T= ½ (n² + n) (χ +1/3 (n-1) (a+1))

In conclusion I state that the formula that I have found works for the step stair, and grid. Therefore it is useful to have theses formulas because it makes problem solve in an efficient way, therefore it allows the problem to be solved in the quickest way.

Ms Akbar

Maths

...read more.

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