I have been asked to find out how many squares will be needed

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

In these drawings I have noticed that if we look at the symmetrical sides of the pattern and then add up the number of squares we can get a square number.

From it.

No of sequence

Black squares

White squares

Total

1st difference

2nd difference

1

1

0

1

4

4

2

1

4

5

8

4

3

5

8

13

12

4

4

13

12

25

16

4

5

25

16

41

20

4

6

41

20

61

24

4

7

61

24

85

28

4

8

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.

N = the number of the sequence

1, from nth term we can see it goes up in fours so this means its 4n.

2, from 4n if we want to get the nth term we have to -4 each time.

This is how I found the formula.

To try to see if the formula works we can substitute it in any sequence.

Example 1: To substitute it in sequence 1 we do the following:

1) 4 x 1=4

2) 4 – 4 = 0

This means the formula is right because nth tern is 0.

Example 2:

1) 4 x 2 = 8

2) 8 – 4 = 4

This is right as well.

Example: 3

1) 4 x 3 = 12

2) 12 -4 = 8

This is right as well.

So formula is right.

This is another table to help me work out the formula for black squares.

No of sequence

1

2

3

4

5

2n�

2

8

18

32

50

– 2n

-2

-4

-6

-8

-10

+1

1

1

1

1

1

Total

1

5

13

25

41

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

The formula for black squares is 2n� – 2n+1.

From the table we can see that the total we got matches the result table total.

To try seeing if the formula works we can substitute it in any of the sequence.

Example: 1

To try it on the first sequence we do the following:

1) 2 x 1� = 2

2) – 2 x1 = -2

3) For this you just add 1

4) Add these up

5) Total is 2+-2+1=1

So the formula works.

Example: 2

To try it on the second sequence we do the following:

1) 2 x 2� = 8

2) -2 x 2 = -4

3) Just add 1

4) Add these up

5) Total is 8+-4+1=5

So the total from the result table shows 5

So the black formula is 2n�-2n+1

From this I have found out that the 1st difference is a linear sequence.

The second difference is a constant and a quadratic sequence an� + 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.