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  • Level: GCSE
  • Subject: Maths
  • Word count: 1115

Number Stairs.

Extracts from this document...

Introduction

  Number Stairs

Introduction

        In this project I will investigate the relationship between the stair total and the position of the stair shape on the grid for three step stairs. Then I will move on to investigating the relationship between the stair totals and other step stairs on other number grids. Below is a 10x10 grid with a three step stair highlighted in yellow and another in red.

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

16

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

Strategy

First we must define the variable. We will first choose the number in the bottom left hand corner of the step stair. I will call it n. I will then move the stair sequence systematically, with n increasing in tens, starting from five. This is because it is impossible to have a three step stair where n is any of the non-highlighted numbers below.

Table of results

n

5

15

25

35

45

55

65

75

total

74

134

194

254

314

374

434

494

...read more.

Middle

We can now put these numbers into the formula and find that

6n+44

is the formula to find the sum of any three step stair on a 10x10 grid.

        To prove this I will put the algebraic values for each square in the three 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

n+20

34

35

36

37

38

39

40

21

22

n+10

n+11

25

26

27

28

29

30

11

12

n

n+1

n+2

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

We can now see that there are six squares in the stair (6n) and the numbers, which show how much they increase n by, add up to 44 (+44)

Step 2

Now we can begin to either vary the size of the grid or vary the size of the stair shape.

        I will vary the size of the grid first because this is the easier option as there will still only be six numbers to add up in the three step stair.

...read more.

Conclusion

Total = as^3 + bs^2 + cs

where s = The number of steps on the stair

In order to work out this formula a simeltaneous equation must be used.

Firstly I will take the equations used for the 2 and 3 step stair.  

3n + 11         6n + 44

These equations can then be manipulated and added to the formula, and then be worked out.

11 = 8a + 4b + 2c             * 3

44 = 27a + 9b + 3c           *2

33 = 24a + 12b + 6c

                                            _

88 = 54a + 18b + 6c

55 = 30a + 6b

55/30 = 1 5/6

From here we can then work out the value of c

11 = 8a + 4b + 2c               * 27

44 = 27a + 9b +2c              *8

297= 216a + 108b + 54c

                                        -

352 = 216a + 72b + 24c

-55 = 36b + 30c

-55/30 = -1 5/6

To test to see if the formula works I will first convert the fractions to top heavy ones.

 1 5/6 = 11/6         -1 5/6  = -11/6    

Now the fractions will be placed into the formula.

(11/6*3^3)   +  (-11/6*3)     = 44.

...read more.

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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