# Maths Sequences Investigation

Extracts from this document...

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

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:

Conclusion

N2 + 8N + 1 N=6

(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

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

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