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

# &amp;#147;C&amp;#148; Totals Investigation.

Extracts from this document...

Introduction

## “C” TOTALS INVESTIGATION

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”).

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 38 39 40 41 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

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.

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

+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:

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

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