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In this assignment I am going to try to find the relation between the t-total and the t-number and then will express this in an algebraic form. I have been asked in the question to find the relationship between the t-total ad the t-number in a nine by

Extracts from this document...

Introduction

Abdul Khan W3                                                                                   16/06/01

T-SHAPES

Introduction

This assignment is called ‘T-Shapes’. In this assignment I am going to try to find the relation between the t-total and the t-number and then will express this in an algebraic form. I have been asked in the question to find the relationship between the t-total ad the t-number in a ‘nine by nine’ grid; I will d this by creating a table for the t-total and t-number. Hence I will try to discover the common difference and then fid the formula connecting the t-number to the t-total.

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        75        76        78        79        80        81

‘T-TOTAL’ Add all the numbers up including the t-number.image00.png

50        51        52

60

‘T-NUMBER’ The number at the bottom of the ‘T’                     image01.png

69                             algebraically Classified an ‘n’                    

I am investigating the relationship between the t-total and the t-number.

image12.png

image23.png

37

20

42

21

47

22

52

23

57

24

62

25

67

26

72

27

77

28

82

29

...read more.

Middle

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        82        83        84        85        86        87        88        89        90

91        92        93        94        95        96        97        98        99        100

image32.png

image02.png

40

22

45

23

50

24

55

25

60

26

65

27

70

28

As before I noticed that the pattern increased by 5 each time the increase by 1. As the common difference is 5;

There must be a ‘5n’ in the formula.

5*22 = 110        so        110-70=40

I have discovered a rule, which is 5n-70

Prediction: Using a 10*10 grid, I predict for the T-total will be 155 when the T-number is 45

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

(24+25+26+35+45) = 155

(5*45)-70 = 155

image03.png

image04.png

7*7

5n-77

8*8

5n-70

9*9

5n-63

10*10

5n-56

11*11

5n-49

12*12

5n-42

I can see that there is a difference of 7

I will now try to find a formula, which relates the T-total and T-number in ANY grid size, and I will also make the T-total and T-number relate with the grid size to find an overall general formula.

T-number        = n

Grid size         = g

(10*10 grid)

n-2gimage05.png

                1        2        3image07.pngimage06.png

n-2g-1

                        12                             n-2g+1image08.png

n-g22image09.png

n

   n

   n-g

   n-2g

...read more.

Conclusion

I know that; As 5 is the most common difference the must be a ‘5n’ in the formula.

27

                        34

                40        41        42

Un = 5n+49

T-Total        = (27+34+40+41+42)= 184

T-Number        =27

My formula 5n+49 or 5*t-number+49 works

Proof of Formula:

                                             nimage15.png

                        2

      n+7                image16.png

                        9                         n+14image17.png

                15        16        17

image19.pngimage18.png

                                                        n+15

n+13

   n

   n+7

   n+13

   n+14

+ n+15

-----------

 5n+49

 I will now try to find a formula, which relates the T-total and T-number in ANY grid size, and I will also make the T-total and T-number relate with the grid size to find an overall general formula.

T-number        = n

Grid size         = g

(10*10 grid)

nimage20.png

2

n+gimage21.png

                        12                    n+2gimage22.png

        21        22        23                          n+2g+1image24.png

image25.png

n+2g-1

   n

   n+g

   n+2g

   n+2g-1

+ n+2g+1

--------------

  5n+7g

So ‘5n+7g’ is the general formula, which works on any grid of any size connecting the T-total, T-number, and the grid size.

Ж▫●₪ 4◊8◊2 ₪●▫Ж                ◊▪◊« ∆ β Ż »◊▪◊image26.png

image33.png

image27.png

...read more.

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