• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  • Level: GCSE
  • Subject: Maths
  • Word count: 2734

Number stairsMy aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step

Extracts from this document...

Introduction

Jaie Wilde

Number stairs

My aim is to investigate the relationship between the stair total and the position of the stair shape on the grid for 3 step stairs and to investigate further the relationship between the stair totals and other step stairs on other number grids.

91

92

93

94

95

96

97

98

99

100image00.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

To begin I will try to identify a pattern. Taking my example I will take all the three stair steps on the bottom row and see if there are any similarities between the results.

  1. 1+2+3+11+12+21=50
  2. 2+3+4+12+13+22=56
  3. 3+4+5+13+14+23=62
  4. 4+5+6+14+15+24=68
  5. 5+6+7+15+16+25=74
  6. 6+7+8+16+17+26=80
  7. 7+8+9+17+18+27=86
  8. 8+9+10+18+19+28=92

Therefore I can now find a formula. The general term for an arithmetic sequence is Un=ab+c. The terms go up in sixes and this tells me that the nth term will include 6 lots of n or 6n.  For the first term n=1, so 6n=6. But the first term is 50 which is 44 more than 6n. This suggests that the formula is 6n+44.

Trying a few values of n will help prove that my formula is correct.

(6 multiplied by 1) +44 =50.                 image08.png

(6 multiplied by 2) +44 =56.

(6 multiplied by 3) +44 =62.

(6 multiplied by 4) +44 =68.

(6 multiplied by 5) +44 =72.

I will now pick a random 3 step stair and test my formula.

76

66

67

56

57

58

Step 1) Locate n which is in bottom left corner.

N=56

Step 2) Insert n into formula 6n+44.

Step3) (6 multiplied by 56) +44= 380 which equals the sum of the numbers.

The formula has worked.

With this formula I can now identify the sum of any 3 step stairs. I will now go on to try and identify similar formulas for bigger stair steps.

...read more.

Middle

I will now test my formula by selecting a random 6 step stair.

image14.png

72

62

63

52

53

54

42

43

44

45

32

33

34

35

36

22

23

24

25

26

27

I have now successfully found formulas for 3, 4, 5 and 6 step stairs.

Number of stairs

formula

3

6n+44

4

10n+110

5

15n+220

6

21n+385

From this table I have noticed that the numbers before n are triangular. Therefore I assume that the formula for a seven step stair will include 28n.

I will now test this on a seven step stair to see if this is true.

61

51

52

41

42

43

31

32

33

34

21

22

23

24

25

11

12

13

14

15

16

1

2

3

4

5

6

7

Also I have noticed that the number of stairs corresponds to the first term of the formula.

3

6n+44

4

10n+110

5

15n+220

6

21n+385

To get from 3 to 6 you multiply by 2.

To get from 4 to 10 you multiply by 2.5.

To get from 5 to 15 you multiply by 3.

To get from 6 to 21 you multiply by 3.5.

If I put this in a table:

1

2

3

4

5

6

1

1.5

2

2.5

3

3.5

The difference between the terms is ½ so my formula must start with ½ n. If I take term 4 I will half it to get two then I will add half to get to 2.5. This applies to all the terms. Therefore my formula is ½ n+ 0.5. This shows me what to multiply the number of step stairs by to find out the first term of the formula.

I will take an 8 step stair as an example to prove my formula.

If I put 8 in the formula it would be ½ of 8 which is 4. Then add 0.5 which is 4.5. This shows me that I have to multiply 8 by 4.

...read more.

Conclusion

25image09.png

19

20

13

14

15

7

8

9

10

1

2

3

4

5

6 step stairs.

57

58

59

60

61

62

63

64image04.png

49

50

51

52

53

54

55

56

41

42

43

44

45

46

47

48

33

34

35

36

37

38

39

40

25

26

27

28

29

30

31

32

17

18

19

20

21

22

23

24

9

10

11

12

13

14

15

16

1

2

3

4

5

6

7

8

50

42

43

34

35

36

26

27

28

29

18

19

20

21

22

10

11

12

13

14

15

I will start by taking a random 6 by 6 step stair. I can see that x = 10. If I write this out algebraically I would get: (x) (x + 1) (x+2) (x+3) (x+4) (x+5) (x+g) (x+g+1) (x+g+2) (x+g+3) (x+g+4) (x+2g) (x+2g+1) (x+2g+2) (x+2g+3) (x+3g) (x+3g+1) (x+3g+2) (x+4g) (x+4g+1) (x+5g)

I will now take a random 6 step stair from a 10 by 10 grid.

95

85

86

75

76

77

65

66

67

68

55

56

57

58

59

45

46

47

48

49

50

If I wrote this out algebraically I find that it would be the same as in the 8 by 8 grid being: (x) (x + 1) (x+2) (x+3) (x+4) (x+5) (x+g) (x+g+1) (x+g+2) (x+g+3) (x+g+4) (x+2g) (x+2g+1) (x+2g+2) (x+2g+3) (x+3g) (x+3g+1) (x+3g+2) (x+4g) (x+4g+1) (x+5g).

If I simplify all this I would find that the equation would be 21x + 35g + 35. This represents the equation for a 6 step stair on any size grid.

I will now test the equation on a random 6 step stair from a 6 by 6 grid.

31

25

26

19

20

21

13

14

15

16

7

8

9

10

11

1

2

3

4

5

6

x=1 and g=6. If I substitute these into the formula I will get (21 multiplied by 1) + (35 multiplied by 6) add 35. This will equal 266. If you add the numbers individually you will also get 266 which shows that the equation works.

Here is a table of my results

Number of steps in staircase

Equation

3

6x+4g+4

4

10x+10g+10

5

15x+20g+20

6

21x+35g+35

I will now try and find an equation for any size stair on any size grid. The final product will be an equation like this: (part1)x + (part2)g + (part3).

...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.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Number Stairs, Grids and Sequences essays

  1. For other 3-step stairs, investigate the relationship between the stair total and the position ...

    42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

  2. Number Stairs

    I will then investigate the formulas to see if I can generate a universal formula for any step stair in any position on the 10 by 10 Grid. In this section the symbols for the inputs will be; (s) Number of Steps Of the Stair for example 3-Step Stair(s)=3 (x)

  1. Number stairs

    same logic and method for other grids such as 11 x 11 and 12 x 12 and 13 x 13 etc. I will use the same 3 step-stair approach and I can then use the algebra formula for the 11 x 11 and 12 x 12 Number Grids to find

  2. Maths - number grid

    I will again use algebra to prove my defined difference of 48 is accurate: (r+2)(r+24) - r(r+26) r(r+24)+2(r+24) - r -26r r +24r+2r+48 - r -26r =48 I have now calculated a trend for my 2x2 squares and came to a difference of 12 and a trend for my 3x3 squares and came to a difference of 48.

  1. Number Stairs

    If N = 4, then T = 4 + 5 +6 +14 +15 +24 = 68. The following table shows the stair total (T) depending on the relevant stair number: N T 1 50 2 56 3 62 4 68 5 74 50 56 62 68 74 +6 +6 +6

  2. Maths Grids Totals

    "x" is the number in the top-left corner. The number to the right of it is "x+ (n-1)" because it is "x" added to the length of the square (take away one). Because the square is on a 10 x 10 grid, the formula for the number underneath the square is "x" added to (10 multiplied by "the squares length"-1).

  1. Mathematics - Number Stairs

    66 Suspected formula: T = 6n + 36 Prediction / Test: 6 x 20 + 36 = 156 36 28 29 20 21 22 20 + 21 + 22 + 28 + 29 + 36 = 156 Algebraic Proof: n+16 n+8 n+9 n n+1 n+2 n + (n+1) + (n+2)

  2. 100 Number Grid

    640 150 20 9th Term 10 x 10 810 170 20 Looking closely at this table, I can see a formula for finding the nth term (n). Multiplying 10 by n2 will provide the product difference for any nth term in a 10 x 10 number grid.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work