• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

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

...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.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE T-Total essays

  1. T-Total Maths

    T = 5N-63 N = 20 21 22 23 T = 37 42 47 52 T-number T = (5x20)-63 = 100-63 T = 37 T-total This equation has produced its first correct answer. I will carry on and test T-shape I know the T-total for N = 20 T =

  2. Connect 4 - Maths Investigation.

    Using the difference method to form an overall rule I multiply the difference (11) by the length and take away n to get the total for any connect. Premature rule: 11L - n For the 5 x 5: 11 x 5 - 27 Rule for Connect 4 with height 5:

  1. T-Total Maths coursework

    = 185-56 = 129 N = 38 T = (5x38)-56 = 190-56 = 134 N = 63 T = (5x63)-56 = 315-56 = 259 Here is the last equation I will show from the 8 by 8 grid to show that the equation is correct.

  2. T-total Investigation

    the T is going up in 10's because I am using the 10by10 grid. With the T set out like this I can see if my formula is correct. I added all the expressions inside the T; this is done in the working below: T-total = T - 22 +

  1. Black and white squares

    Pattern6 61-20 = Pattern5 25+16 Pattern5 41-16 = Pattern4 12+13 My third observation, concerns finding the number of white squares in a pattern, which is found when we take away the total number of squares of a pattern from the total number of squares of the pattern prior to it.

  2. Urban Settlements have much greater accessibility than rural settlements. Is this so?

    The bus network from Bexley extends to Bexleyheath (another bus route centre), Eltham, Bluewater and Dartford. Because the roads through Bexley are very busy, it is sometimes dangerous to attempt to cross, which may also be a factor in the lack of pedestrians.

  1. T-Shapes Coursework

    Section 3, but is repeated for clarification here: [Sum of Wing] = = = = = = = = = (n - p) + ... + (n - 2) + (n - 1) + n + (n + 1) + (n + 2)

  2. T-Shapes Coursework

    If we simplify this equation, we can find the general formula that might apply to any T-Shape rotated 270 degrees clockwise. Tt = n + (n - 1) + (n - 2) + (n + (g - 2)) + (n - (g + 2))

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work