# 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

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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.

50 51 52

60

‘T-NUMBER’ The number at the bottom of the ‘T’

69 algebraically Classified an ‘n’

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

37 | 20 |

42 | 21 |

47 | 22 |

52 | 23 |

57 | 24 |

62 | 25 |

67 | 26 |

72 | 27 |

77 | 28 |

82 | 29 |

Middle

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

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

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-2g

1 2 3

n-2g-1

12 n-2g+1

n-g22

n

n

n-g

n-2g

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:

n

2

n+7

9 n+14

15 16 17

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)

n

2

n+g

12 n+2g

21 22 23 n+2g+1

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 »◊▪◊

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