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have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence

Extracts from this document...

Introduction

Page 1

Introduction

I have been asked to find out how many squares would be needed to make up a certain pattern according to its sequence. In this investigation I will be aiming to find a formula which could be used to find out the number of squares needed to build the pattern at any sequential position. Firstly I will break the problem down into simple steps to begin with and go into more detail to explain my solutions such as the nth term.

In order to find this I would need to work of the formula:

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    Term 1                         Term 2                               Term 3                                                 Term 4

       B=1                            B=5                                    B=13                                                    B=25

      W=4                           W=8                                   W=12                                                   W=16

Pattern

Dark squares

White squares

Number of     squares

1

1

4

1 + 4 = 5

2

5

8

5 + 8 = 13

3

13

12

13 + 12 = 25

Hypothesis

I predict that there will be a difference between each cross-section shape.

Therefore there will be a quadratic sequence which can tell me how many squares would be needed in any given shape.

Page2

Prediction

1 + 3 + 1 = 5

1 + 3 + 5 +3 + 1 = 13

1 + 3 + 5 + 7 + 5 + 3 + 1 = 25

1 + 3 + 5 + 7 + 9 + 7 + 5 + 3 + 1 = 41

...read more.

Middle

181

9

181

21 + 19 =40

221

Page 3

Differences

Total

 5     13      25      41     61      85      113     145   181    221

1st difference

    4     8      12      16      20      24      28      32      36      40

2nd difference

        4       4       4       4        4        4        4        4             

This shows a main difference of 4. I think this will influence the formula. I think this will mainly be in the form of a multiple of 4.

The first formula I will try to find is the formula for the surrounding white squares.

Finding a formula for the number of squares

Term

1

2

3

4

No. of squares

5

13

25

41

1st difference

8

12

16

20

2nd difference

4

4

4

4

2n2

2

8

18

32

Rest of sequence

3

5

7

9

Final difference

2

2

2

2

2n2+2n+1

e.g.

n=2

2 x 22 + 2 x 2 + 1=13

Page 4

Trying for a formula – white squares.

In each case I have observed that if you multiply the pattern number by 4 it gives you the amount of white squares.

E.g. Pattern

1 x 4 = 4 white squares

2 x 4 = 8 white squares

3 x 4 = 12 white squares

4 x 4 = 16 white squares

This goes on & by using this method you can find the amount of white squares as long as you have the pattern number.

...read more.

Conclusion

The differences between the dark & total number of squares once again go up in 4.

I think to find a formula for the dark squares you can find a possible formula to find the amount of total number of squares & then minus the formula for the white squares.

Page6

Term

1

2

3

4

No. of dark squares

1

    5

      13

25

1st difference

4

8

12

16

2nd difference

4

4

4

4

2n2

2

8

18

32

Rest of sequence

-1

-3

-5

-7

Final difference

-2

-2

-2

-2

2n2–2n + 1

e.g.

n=2

2 x 22 – 2 x 2 + 1= 5

Conclusion

In conclusion I found made a hypothesis to predict what I though of the differences in the cross-section shapes, this helped me to achieve the formula for the squares changes, the white squares changes and the dark squares changes, I also used tables and algebra sequences to help me find the formulas, they turned out to be very useful in helping me with my investigation.

...read more.

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