Investigation of a Relationship Between Stairs

Investigation of a Relationship Between Stairs

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Investigation of a Relationship Between Stairs

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

The total of the numbers in side the stair is

25+26+27+35+36+37+45=194


I will start on a 2-step stair
------------------------------

38 28+29+38=95

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Related Searches


28 29

37 37+27+28=92

27 28

36 36+26+27=89

26 27

I have found out that every time you move the two-step stair one to
the left the product of the two-step stair goes down by three. This
also means that every time you move the two-step stair one to the
right the product will go up by three.

23 23+13+14=50

13 14

33 33+23+24=80

23 24

43 43+33+34=110

33 34

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

35 36

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

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

62 63 64

81

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

61 62 63

80

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

60 61 62

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. This
also means that every time you move the three-step stair to the right
the product will go up by six.

33

23 24 33+24+23+15+14+13=122

13 14 15

43

33 34 43+33+34+23+24+25=182

23 24 25

53

43 44 33+34+35+43+44+53=242

33 34 35

The product of the last three three-step stairs tells me that when you
move the stairs up by one the product of the three-step stair goes up
by sixty. This also means that when you move the three-step stair down
by one the product will go down by -60. I have also found out the nth
term for the three-step stairs and it is 6n+44 I shall now test this.

78

68 69 58+59+60+68+69+78=392

58 59 60

To do the nth term I would have to times 6 by 58 then plus 44 which
equals 392. Now I will move on to a four-step stair.

I shall now move on to a four-step stair

83

73 74

63 64 65 53+54+55+56+63+64+65+73+74+83=640

53 54 55 56

82

72 73

62 63 64 52+53+54+55+62+63+64+72+73+82=630

52 53 54 55

81

71 72

61 62 63 51+52+53+54+61+62+63+71+72+81=620

51 52 53 54

I have found out that every time you move the four-step stair one to
the left the product of the four-step stair goes down by ten. This
also means that every time you move the four-step stair to the right
the product will go up by ten. I have also found out that all the
products are even and I know that they will be even wherever you put a
three or four-step stair it will always be even. On a three step stair
wherever you put it on the grid there will always be 2 or 4 odd
numbers and if you add 2 odds numbers together it makes a even and the
same with 4 odd numbers and with a four-step stair there is ever 4 or
6 odd numbers.

43

33 34

23 24 25 13+14+15+16+23+24+25+33+34+43=240

13 14 15 16

53

43 44

33 34 35 23+24+25+26+33+34+35+43+44+53=340

23 24 25 26

63

53 54

43 44 45 33+34+35+36+43+44+45+53+54+63=440

33 34 35 36

I have found out that the product of the four-step stairs tells me
that when you move the stairs up by one the product of the four-step
stair goes up by one hundred. This also means that when you move the
four-step stair down by one the product will go down by -100. I have
also found the nth term for the four-step stair is 10n+110. I shall
now test the nth term.

56

46 47

36 37 38 26+27+28+29+36+37+38+46+47+56=370

26 27 28 29

To do the nth term I have to times 10 by 26 then plus 110 which equals
370.

So I will now move on to a five-step stair.

83

73 74

63 64 65 43+44+45+46+47+53+54+55+56+63+64+65+73+74+83=865

53 54 55 56

43 44 45 46 47

82

72 73

62 63 64 42+43+44+45+46+52+53+54+55+62+63+64+72+73+82=850

52 53 54 55

42 43 44 45 46

81

71 72

61 62 63

51 52 53 54 41+42+43+44+45+51+52+53+54+61+62+63+71+72+81=835

41 42 43 44 45

I have found out that every time you move the five-step stair one to
the left the product of the three-step stair goes down by fifteen.
This also means that every time you move the three-step stair to the
right the product will go up by fifteen, but the product is now not
always an even number it is sometimes an even or an odd number. This
is because there is not always an even amount of odd numbers, there is
some times an odd amount of odd numbers. I predict that when you move
a five-step stair up by one the product will go up by 150. This is
because when you move it up each number is going up by 10 and there is
15 numbers in the number so this means that it would go up by 150.

53

43 44

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

23 24 25 26

13 14 15 16 17

63

53 54

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

33 34 35 36

23 24 25 26 27

73

63 64

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

43 44 45 46

33 34 35 36 37

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

54 55

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

34 35 36 37

24 25 26 27 28

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.
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