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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

For the fourth Column (+) the numbers go up in 2’s because when the length is increased by one an extra 2 +’s are added to the middle. The nth term is (n-1) x2.

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

n x 8.

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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