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  • Level: GCSE
  • Subject: Maths
  • Word count: 2191

Nth Term Investigation

Extracts from this document...

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 nimage00.pngimage00.pngimage01.png

+

1 x 1

4

0

0

2 x 2

4

4

1

3 x 3

4

8

4

4 x 4

4

12

9

image01.png

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= 4image00.pngimage02.pngimage00.png

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.  image02.pngimage03.pngimage03.pngimage00.pngimage03.png

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 nimage00.pngimage00.pngimage00.png

+

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

...read more.

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 timage00.pngimage00.pngimage00.pngimage02.png

+

10 x 3

4

22

18

25 x 3

4

52

48

50 x 3image04.png

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 timage00.pngimage02.pngimage00.pngimage01.png

+

1 x 4

4

6

0

2 x 4

4

8

3

3 x 4image04.png

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= 4image02.pngimage00.png

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.image02.pngimage00.pngimage03.pngimage02.png

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 timage00.pngimage00.pngimage01.pngimage00.pngimage02.png

+

10 x 4

4

24

27

25 x 4

4

54

72

50 x 4image04.png

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)

...read more.

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

...read more.

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