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• Level: GCSE
• Subject: Maths
• Word count: 1063

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

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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:

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.

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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

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.

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.

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 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.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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