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  • Level: GCSE
  • Subject: Maths
  • Word count: 1322

“C” Totals Investigation.

Extracts from this document...

Introduction

“C” TOTALS INVESTIGATION

WHAT IS THE INVESTIGATION ABOUT?

In this investigation, I have tried to experiment using different grids but the same C shape.

WHAT IS THE PROBLEM?

I am going to start with a 9x9 grid with 81 counting numbers in it and write them down. I will then draw a as shown below in my mini diagram:

The bottom right number is going to be my C number (n) and the total of all the numbers in the C is going to be my C total, (I am going to call this “t”).

image00.png

image01.png

Finally, I am going to see if there is a relationship between “n” & “t”.

MY GRID…

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

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40

41

42

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45

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47

48

49

50

51

52

53

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59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

Below I have drawn some random C’s, which I took from the table.

41

42

50

59

60

N=60

T=60+50+59+41+42=

        252

8

9

17

26

27

N=27

T=8+9+17+26+27=

        87

56

57

65

74

75

N=75

T=56+57+65+74+75=

        56

57+

                           113

65+

                           178

149+

327

3

4

12

21

22

N=22

T=3+4+12+21+22=

        62

From above I can see a pattern forming; the T numbers are ending in a 2 or a 7:

252, 87, 327, and 62.

...read more.

Middle

22

23

N=23

T=4+5+13+22+23=

        67

5

6

14

23

24

N=24

T=5+6+14+23+24=

        72

6

7

15

24

25

N=25

T=6+7+15+24+25=

        77

A TABLE SHOWING MY RESULTS…

N

20

21

22

23

24

25

T

52

57

62

67

72

77

image02.pngimage02.pngimage02.pngimage02.png

        +5                   +5                +5        +5                +5

I can see that my prediction was right because all the numbers go up in 5’s.

THE FORMULA…

I have already established from the table that the formula will have 5 in it somewhere, this is because the numbers as shown in the table go up in 5’s.

The start of my formula is going to be like this:

T=5n-?? ~(?? =The 0th term)

Now I have to find the 0th term.

To do this I am going to multiply 20 by 5 giving us 100 then I am going to subtract 52-the 20th term from 100 giving me the 0th term, which is: -48

0th

20th

-48

52

So now, I can complete my formula:

T=5n-48

EXPERIMENTATION WITH C’S…

Now I am going to translate the C downwards.

1

2

10

19

20

N=20

T=1+2+10+19+20=

        52

Translating the C downward by one square…

10

11

19

28

29

N=29

T=10+11+19+28+29=

         97

All the numbers seem to be in relation with the number 9:

~97-52=45

  45\5=9

~RED=

 10-1=9

~YELLOW=

11-2=9

~GREEN=

  19-10=9

~BLUE=

28-19=9

~PINK=

29-20=9

I think that the number 9 keeps appearing is because the grid, which, I have drawn, is a 9x9 grid.

Below is a C like the other one’s but with n terms in it instead of the numbers from the grid.

N-19

N-18

N-10

N-1

N

From this I discovered that wherever in the 9x9 grid you go the N terms will always be the same. Though I do not think the C can be rotated etc.

Below I am going to draw a 9x9 grid again but this time I am going to alter the way I draw the C’s. This time I am going to put the C on its side like shown below:

...read more.

Conclusion

N=3

3*5=15+22

=T=37

EXPERIMENTING WITH GRIDS OF DIFFERENT SIZES…

Below I am going to draw a 4x4 grid, in which, like the 9x9 grid I am going to experiment with.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Now I am going to take some C’s and experiment with them

1

2

5

9

10

N=10

T=1+2+5+9+10=

        27

2

3

6

10

11

N=11

T=2+3+6+10+11=

        32

3

4

7

11

12

N=12

T=3+4+7+11+12=

        37

5

6

9

13

14

N=14

T=5+6+9+13+14=

        47

Because the grid is too small for me to see what the T number is going to be when the N number is 13. However, I assume it is going to be 42.

A table with my results…

N

10

11

12

13

14

T

27

32

37

42

47

The Formula is going to have the number 5 in it, because the T numbers go up in 5’s:

T=5n????

10*5=50

27-50=-23

Therefore the formula is:

T=5n-23

To check this:

N=10

5*10=50-23

=T=27

1

2

5

9

10

N=10

T=10+9+1+2+5=

        27

5

6

9

13

14

N=14

T=5+6+9+13+14=

        47

All the numbers, like the 9x9 grid, seem to be associated with the number 4.

~47-27=20

20/5=4

~ 5-1=4

~ 6-2=4

~ 9-5=4

~ 13-9=4

~ 14-10=4

I think that the number 4 keeps appearing is because the grid, which, I have drawn, is a 4x4 grid.

Mehjabeen Iqbal 10NDM        Page         5/10/2007

...read more.

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