# T total and t number

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Introduction

## GCSE MATHEMATICS COURSEWORK

## Introduction

My aim is to investigate the relationship between the T-total and the T-number.

I will then used grids of different sizes and translated the T-shape to different positions to re-investigate the relationship between the T-total and the T-number.

###### Figure 1.Figure 1.a.

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 |

(T-19) 1 | (T-18) 2 | (T-17) 3 | ||||||

10 | (T-9) 11 | 12 | ||||||

19 | (T) 20 | 21 |

We shall call the grid dimensions ‘g’. Figure 1is drawn on a 9 by 9 grid, so in this case: g = 81

Figure 1 is a T-shape drawn on a number grid where g = 9.

### The T-total is 1+2+3+11+20=37

The T-number is 20.

The Nth term, as taken from figure 1.a. is:

T + (T-9) + (T-17) + (T-18) + (T-19) = 5T – 63

5T – 63 = 37

5T = 37 + 63

T = 100

5

T = 20

Figure 2.

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 |

### In this grid g = 9. The T-total is 2+3+4+12+21=42

### The T number is 21

The Nth term is will be the same:

T + (T-9) + (T-17) + (T-18) + (T-19) = 5T – 63

5T – 63 = 42

5T = 42 + 63

T = 105

5

Middle

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### This time g = 10. The T-total is 1+2+3+12+22=40

The T number is 22

The Nth term is:

T + (T-10) + (T-19) + (T-20) + (T-21) = 5T – 70

5T – 70 = 40

5T = 40 + 70

T = 110

5

T = 22

Figure 5.

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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

### In this grid g = 10. The T-total is 2+3+4+13+23=45

The T number is 23

The Nth term is will be the same:

T + (T-10) + (T-19) + (T-20) + (T-21) = 5T – 70

5T – 70 = 45

5T = 45 + 70

T = 115

5

### T = 23

Figure 6.

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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

### In this grid g = 10. The T-total is 3+4+5+14+24=50

The T number is 24

Conclusion

G = 10 is -70

G = 11 is -77

Note 3:

In note 2 it became apparent that all of the Nth terms had a factor of -7 in them. Therefore, to find the Nth term of any size square grid, you took the number that the grid was multiplied by, then multiplied that number by –7.

ie. When g = 62 you will have a Nth term of -434:

g x –7 = -434

Note 4:

If you just had a Nth term and wanted to trace it back to see what the dimensions of the grid were that it came off you would reverse the sum.

ie. Take the Nth term, divide it by –7 to get the dimensions of the square grid:

-511

-7 = 73

A Nth term of –511 came from a 73 x 73 square grid.

CONCLUSION

As we have found out, the pattern for The Nth term will be shown by the formula:

5T – 7g

T-2g-1 | T-2g | T-2g+1 |

T-g | ||

T |

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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