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Maths Sequences Investigation

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Introduction

Maths Coursework

        Firstly I drew out a series of patterns so I could study the many different aspects of them. Here is what I found:

Black Squares

White Squares

Diameter

Total Squares

1

4

3

5

5

8

5

13

13

12

7

25

25

16

9

41

41

20

11

61

61

24

13

85

85

28

15

113

113

32

17

145

  • The number of Black Squares increases each time by the number of white squares.
  • The number of White squares increases in multiples of four every time
  • The total squares equals the sum of Black Squares and White Squares
  • The diameter increases by two every time

Because this sequence of patterns follows a sequence and increases by some

...read more.

Middle

13

85

7

85

28

15

113

8

113

32

17

145

Now I can use n as a basis for building equations used to define the different aspects such as the ones in the table above. Here is the equation for the number of white squares:

4 x N = the number of white squares

4N = W

And I also worked out the equation for the Diameter of any pattern:

2 x N + 1 = the diameter of the pattern

2N+1 = D

        These two equations were relatively easy to work out because they follow a set increment. I simply looked down the table above and noted down the difference in them:

White Squares

Difference

Diameter

Difference

4

3

4

2

8

5

4

2

12

7

4

2

16

9

4

2

20

11

4

2

24

13

4

2

28

15

4

2

32

17


Forming an equation for the total number of squares is slightly harder. Now I will explain how I worked it out:

...read more.

Conclusion

N2 + 8N + 1                N=6

image05.png

(6 x 6) + (8 x 6) + 1   =   36 + 48 + 1   =   85  


Another Prediction:

N2 + 10N + 1                N=8

(8 x 8) + (10 x 8) + 1   =   64 + 80 + 1   =   145   Correct

Now I will add this to my equation:

N2 + 6N + 1

(Incorrect Version)

I need to change to coefficient to reflect ‘N + 2’

N2 + N(N+2) + 1

This is now correct but can be simplified:

N2 + N2 + 2N +1

And further:

2N2 + 2N + 1


This is now my final formula. I am going to test it on the same patterns I did before just in case I have made an error.

N=6                2N2 + 2N + 1        = 2(6 x 6) + (2 x 6) + 1

= (2 x 36) + 12 + 1

= 72 + 12 + 1

= 85                 Correct

N=8                2N2 + 2N + 1        = 2(8 x 8) + (2 x 8) + 1

                                        = (2 x 64) + 16 + 1

                                        = 128 + 16 +1

                                        = 145                 Correct

Now I know this formula works, I am able to predict the number of squares in much bigger patterns:

N=20                2N2 + 2N + 1        = 2(20 X 20) + (2 X 20) + 1

                                        = (2 X 400) + 40 + 1

                                        = 800 + 40 + 1

                                        = 841 total squares

...read more.

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