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In this course work I have been asked to find out how many squares will be needed to make up a certain pattern according to its sequence.

Extracts from this document...

Introduction

G.C.S.E MATHEMATICS BORDERS COURSEWORK

VIPUL VASANT11ROKingsbury high school        p        

INTRODUCTION

In this course work I have been asked to find out how many squares will be needed to make up a certain pattern according to its sequence.

I hope to find a formula that could be used to find the number of squares needed to build the next pattern at any of the sequences. I will carry out this experiment on a 2d pattern and then try doing it on 3d pattern.

1st sequence

1 black square        

0 white square        2nd sequence

        1 black square

        4 white squares        3rd sequence

        5 black squares

         8 white squares

4th sequence

13 black squares         

12 white squares        5th sequence

        25 black squares

        16 white squares

6th sequence

41 black squares                7th sequence

20 white squares        61 black squares

        24 white squares

8th sequence

85 black squares

28 white squares

9th sequence

        113 black squares        

        32 white squares

...read more.

Middle

        85

        28

      113

        32

       4

           9

      113

        32

      14 5

        36

       4

         10

      145

        36

      181

        40

       4

         11

      181

        40

      221

       44

       4

            12

      221

        44

      265

       4

Table showing my results.

I have achieved these results in the table I have shown above. I got these results from the drawings I have drawn.

From the results we can see that there are a lot of patterns.

I can see that the total has all the odd numbers in it.

Another pattern I can see is the black squares has the same numbers as the total but first that number comes on total then on the black squares.

Example: if you look at sequence no.4 you can see there are 13 black squares and then the same number comes on sequence 3 but this time on the total column.

Table to help me work out the formula for white squares.

No of sequence

1

2

3

4

5

Nth term

0

4

8

12

16

4N

4

8

12

16

20

From the table we can find the formula for white squares.

The formula for white squares is 4N-4.

...read more.

Conclusion

 difference is a linear sequence.

The second difference is a constant and a quadratic sequence an² + bn +c.

This is because it always goes up in fours.

I am going to extend the investigation to 3 dimensions.

These are 3d patterns.

Sequence 1

1 Cube

Sequence 2

7 cubes

Sequence 3

In this sequence there are 25 cubes.

There are 6 lines each with four cubes in a line which means

6 x 4 = 24.

Also adding the middle one you get 1 more so total is 25 cubes

No. of sequence

No. of cubes

1st difference

2nd difference

3 rd difference.

1

           1

 6

12

8

2

           7

18

20

8

3

          25

38

28

8

4

          63

66

36

8

5

        129

102

44

8

6

        231

            146

52

8

7

        377

            198

60

8

        575

            258

9

        833

This is a table showing the number of sequence in 3d pattern.

        From the drawings I was able to make this table and from the differences I was able to find out the number of cubes all the way up to sequence 9.  

The arrows show the differences between the numbers.

From the difference I got the second difference then I got the first difference. This made it easy for me and then I got the number of cubes.

...read more.

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

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