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  • Level: GCSE
  • Subject: Maths
  • Word count: 2779

Number Stairs.

Extracts from this document...

Introduction

Katie Muston                                                                          

Number Stairs

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

On the 10 by 10 number grid above I have drawn a stair shape. This is known as a three-step stair.

The total of the numbers inside the stair is:

                                55+45+46+35+36+37=254

So the stair total for this shape is 254.

The stair number for this shape is stair 35 as the number in the bottom left hand corner is 35.

        Now we can investigate the relationship between the position of the stair shape on the grid and the stair total. The diagram below is stair 1 as the number in the bottom left hand corner is 1.

21

11

12

1

2

3

As we see, the stair total for this 3-step stair1 on this 10 by 10-number grid is 50 as 1 + 2 + 3 + 11 + 12 + 21 = 50.

Below is the stair shape one square to the right of stair1. This stair is called stair2 as the number in the bottom left hand corner is 1. Stair2 is a translation of stair1 one square to the right.

22

12

13

2

3

4

The stair total for this stair shape is 2 + 3+ 4 + 12 + 13 + 22 = 56.

Now we are going to find the step total for stair3 (a translation of stair2 one step to the right).

23

13

14

3

4

5

The stair total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62.

Now we can find the stair total for stair4 (which is a translation of stair 3 one square to the right).

24

14

15

4

5

6

...read more.

Middle

Stair Number:  1                         11                        21                        31

Stair total:      50                          110                        170                        230                

Difference:                    60                         60                               60

I can now predict that the stair total for stair 41 is going to be 230 + 60=290.

Here is the stair to prove my prediction is correct.

61

51

52

41

42

43

The stair total for this shape is 41 + 42 + 43 + 51 + 52 + 61 = 290.

We can therefore say that every time you move the shape one square down the total decreases by 60 and when you move one square up the grid then total increases by 60. Also when you move one square to the right of the grid the total increases by 6 and when you move one square to the left the total decreases by 6.

The reason the total increases by 60 when you move the shape one square up on the grid is because since there are 6 squares in a 3-step stair and each separate number increases by 10. 6 multiplied by 10 equals 60.

The reason the total increases by 6 when you move the shape one square to the right on the grid is because there are 6 squares in a 3-step stair and each separate number increases by 1. 1 multiplied by 6 equals 6.

We can now introduce algebra in order to find a common formula to find the stair total for and 3-step stair on a 10 by 10-number grid.

x+20

x+10

x+11

x

x+1

x+2

...read more.

Conclusion

 7a+3b+c        ,     9a+5b+c        ,   37a+7b+c        ,         61a+9b+c

           12a+2b        ,      18a+2b        ,    24a+2b

6a        ,        6a        

I am now going to match the numbers in the cubic sequence formula above to the numbers that correspond to their positions in the co-efficient in front of g sequence.

6a = 1

a   = 1/6

12a+2b = 3

12(1/6) + 2b = 3

2 + 2b = 3

2b = 1

b = 0.5

7a+3b+c= 3

7(1/6)+3(1/2)+c = 3

c= 1/3

a+ b + c + d = 1

1/6 + ½ + 21/3 + d = 1

d = 0

Therefore the equation for this part of the sequence will be :

(1/6 n3 + 1/2n2 + 1/3n) = coefficient of g

Section 3 : The Sequence For The Constant

The sequence for the constant will go exactly the same as the sequence for the co-efficient of g as all these values are the same. Therefore the equation for this part of the sequence will be :

(1/6 n3 + 1/2n2 + 1/3n) = the constant

Conclusion

 Now that we have finished the three parts of the sequence we can put them together in order to make one equation for any size step-stair on any size grid.

(1/2 n2  + 3/2n + 1)x + (1/6 n3 + 1/2n2 + 1/3n)g + (1/6 n3 + 1/2n2 + 1/3n)= stair total

        x means the number of squares in the stair shape.

g stands for grid size.

n is the equation number. ( If you don’t know the equation number it is the same number as the number of squares in the 2nd column from the left of the stair shape).

        -  -

...read more.

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