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Investigate the relationship between different stairs placed in different places in a number square, which is numbered from one to one hundred.

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Introduction

Introduction

In this project I am going to investigate the relationship between different stairs placed in different places in a number square, which is numbered from one to one hundred. I will be trying to find a formula to get a total of any stair any place.

Example

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

20

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

This is a 3-step stair

...read more.

Middle

23                23+13+14=50

1314

33                33+23+24=80

2324

43                43+33+34=110

3334

I have found out that when you move the stairs up by one the product of the two-step stair goes up by 30. This also means that when you move the two-step stair down by one the product will go down by -30. I have also found out the nth term for all two-step stairs that 3n+11 will equal the sum off all the numbers in the stair. I found out this by multiply the number in the bottom left hand corner by how many numbers that are in the stairs, then I added how much is needed to make the product correct, I shall use this to try and find out the other nth terms. I will now test my formula.

45                45+35+36=116

3536

To do the nth term I have to times 3 by 35 then plus 11, which equals 116

I will now move on to a three-step stair.

82

7273                82+73+72+64+63+62=416

626364

81

7172                81+72+71+63+62+61=410

616263

80

7071                80+71+70+62+61+60=404  

606162

I have found out that every time you move the three-step stair one to the left the product of the three-step stair goes down by six.

...read more.

Conclusion

53

4344

333435                53+43+44+33+34+35+23+24+25+26+13+14+15+16+17=415

23242526

1314151617

63

5354

434445                63+53+54+43+44+45+33+34+35+36+23+24+25+26+27=565

33343536

2324252627

73

6364

535455                73+63+64+53+54+55+43+44+45+46+33+34+35+36+37=715

43444546

3334353637

I have found out that every time you move the 5-step stair up by 1 the product goes up by 150, this means that when you move the 5-step stair down by one the product goes down by 150. I have also found out the nth term for this which is 15n+220, I shall now test this.

64

5455

444546                64+54+55+44+45+46+34+35+36+37+24+25+26+27+28=580

34353637

2425262728

To do the nth term I have to times 15 by 24 then plus 220 which equals 580. The nth term is correct.

Nth term

How many steps

Difference 1 (how much you plus)

Difference 2 (how much you times)

1n+0

1

3n+11

2

22

2

6n+44

3

33

3

10n+110

4

44

4

15n+220

5

5

With this information I can find out any nth term. I shall now try and find out more nth terms.

Nth term

How many steps

Difference 1

Difference 2

21n+165

6

55

6

28n+231

7

66

7

36n+308

8

77

8

45n+398

9

88

9

55n+495

10

99

10

Conclusion

Overall I have found out the nth terms for a 1 to a ten-step stair and some patterns with the nth term. I have also found out how much a 1 to a 5-step stair product changes if you move it up, down, left and right.

...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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