# Nth Term Investigation

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Introduction

## Introduction

In this bit of coursework I will draw shapes that tessellate and work on the parts in between them, I will look for patterns and will try to find the nth term, write a table of my results and draw line graphs as another way of showing my results.

## Squares

I will firstly be looking at squares.

Here is a table showing my results:

n x n | + | ||

1 x 1 | 4 | 0 | 0 |

2 x 2 | 4 | 4 | 1 |

3 x 3 | 4 | 8 | 4 |

4 x 4 | 4 | 12 | 9 |

For the second column ( ) each square has 4 because the symbol represents the corners and all squares have 4 corners. The nth term is n= 4

For the third column ( ) the amount goes up in 4’s because an extra symbol is needed 1 more time on each of the sides and there is 4 sides. The nth term for this is (n-1) x4.

For the fourth column the numbers are square numbers because in the middle where these (+) are found they are in a formation of a square 2 by 2, 3 by 3 etc so you times them and they are square numbers. The nth for this one is (n-1)2.

Here are my predictions for other squares with different lengths.

n x n | + | ||

10 x 10 | 4 | 36 | 81 |

25 x 25 | 4 | 96 | 576 |

50 x 50 | 4 | 196 | 2401 |

100 x 100 | 4 | 396 | 9801 |

## Rectangles No.1

I will now move

Middle

Here are my predictions for other rectangles with different lengths.

n x t | + | ||

10 x 3 | 4 | 22 | 18 |

25 x 3 | 4 | 52 | 48 |

50 x 3 | 4 | 102 | 98 |

100 x 3 | 4 | 202 | 198 |

Rectangles No.3

I will now do the same as the first and second lot of rectangles but instead of (t) being 2 or 3 it will be 4.

Here is a table showing my results:

n x t | + | ||

1 x 4 | 4 | 6 | 0 |

2 x 4 | 4 | 8 | 3 |

3 x 4 | 4 | 10 | 6 |

4 x 4 | 4 | 12 | 9 |

For the second column ( ) each rectangle has 4 because the symbol represents the corners and all rectangles have 4 corners. The nth term is n= 4

For the third column ( ) the numbers go up by 2’s because the length increases by one and an extra T-shape symbol has to be added to both sides. The nth term is (n+2) x2.

For the fourth column (+)the numbers go up by 3’s because when the length is increased by one an extra 3 +’s are added to the middle. The nth term is (n-1) x3.

Here are my predictions for other rectangles with different lengths.

n x t | + | ||

10 x 4 | 4 | 24 | 27 |

25 x 4 | 4 | 54 | 72 |

50 x 4 | 4 | 104 | 147 |

100 x 4 | 4 | 204 | 297 |

I noticed with the first rectangles the nth term for + was n-1, for the second lot of rectangles it was (n -1)

Conclusion

For the fifth column the numbers go up in ones because an extra symbol is added to the middle when the cuboid size increases. The nth term is (n+1) x4.

Here are my predictions for other cubes with different lengths.

n x n x n | ||||

10 x 3 x 3 | 8 | 52 | 80 | 44 |

25 x 3 x 3 | 8 | 112 | 200 | 104 |

50 x 3 x 3 | 8 | 212 | 400 | 204 |

100 x 3 x 3 | 8 | 412 | 800 | 404 |

Cuboid 3

Another set of cuboids.

Here is a table showing my results:

n x n x n | ||||

1 x 4 x 4 | 8 | 16 | 8 | 0 |

2 x 4 x 4 | 8 | 20 | 16 | 4 |

3 x 4 x 4 | 8 | 24 | 24 | 8 |

4 x 4 x 4 | 8 | 28 | 32 | 12 |

For the second column ( ) each cuboid has 8 because the symbol represents the corners and all cubes have 8 corners. The nth term is n= 8.

For the third column ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (n +5) x4.

For the fourth column the numbers ( ) the numbers go up in 4’s because an extra symbol is needed 1 more time on each of the 4 sides. The nth term for this is (nx12) +6.

For the fifth column the numbers go up in ones because an extra symbol is added to the middle when the cuboid size increases. The nth term is (n-1) x9.

Here are my predictions for other cubes with different lengths.

n x n x n | ||||

10 x 4 x 4 | 8 | 60 | 126 | 99 |

25 x 4 x 4 | 8 | 120 | 306 | 216 |

50 x 4 x 4 | 8 | 220 | 606 | 441 |

100 x 4 x 4 | 8 | 420 | 1206 | 891 |

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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