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This coursework will be to investigate to see how many squares would be needed to make any cross-shape build up in this way.

Extracts from this document...

Introduction

Introduction

The figure below shows a dark cross-shape that has been surrounded by white squares to create a bigger cross- shape:

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image01.pngimage01.pngimage01.pngimage00.pngimage00.png

image01.pngimage00.pngimage01.pngimage00.pngimage01.pngimage02.pngimage01.png

image02.pngimage00.pngimage01.pngimage01.pngimage00.png

image01.pngimage00.pngimage00.png

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The bigger cross- shape consists of 25 small squares in total. The next cross- shape is always made by surrounding the previous cross-shape with small squares.

This coursework will be to investigate to see how many squares would be needed to make any cross-shape build up in this way.

Method

First of all I'm going to work out a sequence which consists of a formula. Then I will draw a maximum of six cross-shapes adding on to each sequence, at the end I will produce one extra to testify the formula.

After creating two tables to work out the first

...read more.

Middle

nd difference turned out to be 4. This is called a quadratic function, this means when it takes two sets of terms to actually find out what the constant difference is.

In the following table I will be intending to find out what the answers are to 2n2.

Table 2

Shape Number

1

2

3

4

5

6

Number of Squares

1

5

13

25

41

61

2n2

2

8

18

32

50

72

Remaining sequence

-1

-3

-5

-7

-9

-11

1st difference

-2

-2

-2

-2

-2

-2

No of squares:

          1                   5                   13                  25                  41                 61image03.pngimage03.pngimage04.pngimage04.pngimage04.pngimage04.pngimage03.pngimage03.pngimage04.pngimage03.png

Remaining Sequence:

                    -1                 -3                    -5                   -7                   -9image03.pngimage03.pngimage03.pngimage04.pngimage04.pngimage03.pngimage04.pngimage04.png

1st Diff:

                             -2                   -2                    -2                    -2

To find the remaining sequence, I used the formula 2n2  with the number of squares . So basically when I worked out the answer to 2n2, I subtracted the number of squares by the results of my formula, which gave me the answers to the remaining sequence.

2          12    = 2image06.pngimage05.png

2          52    =  8image06.pngimage05.png

2          132 =18image05.pngimage06.png

2          252=  32image06.pngimage05.png

2          412= 50image06.pngimage05.png

  1. 612= 72image05.pngimage06.png

...read more.

Conclusion

The 3rd difference is therefore 8 which means:

  1. a= 8image09.pngimage10.png

                       6

The 1st part of the formula becomes 8     nb  =  4    n3image09.png

  1. 3

I will now use 4       n2 to make a sequence.

                              3

n

1

2

3

4

5

6

4

    3  n3

4

     3

32

      3

158

         3

256

          3

500

          3

864

          3

Remaining Sequence

=

1 -  4

            3

-1/3

7 - 32

             3

-11/3

25- 158

                3

-33/3

63 - 256

                3

-67/3

1st Diff

-10

        3

-22

        3

-34

        3

2nd Diff

-12

         3

-12

         3

-12

       3

The second part of the equation will now be:

-6       n2      =      -2n2

      3

I will use this to make another sequence.

n

1

2

3

4

5

6

-2n2

-2

-8

-18

-32

Remaining Sequence

-1/3-(-2)

5/3

-11/3-(-8)

13/3

-33/3-(-18)

21/3

-67/3-(-32)

29/3

37/3

1st Diff

8/3

8/3

8/3

8/3

I have now found out the last part of the formula which is :

8   n    -     3

   3                 3

The formula is :     4/3n3   -   2n2   +   8/3n   -    3/3

                          = 4n3    -     6n    +    8n  -   3

                                                    3

To testify this formula I'm going to draw squares so that the 3D look can be easily understood.

...read more.

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