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  • Level: GCSE
  • Subject: Maths
  • Word count: 1490

Border sequences

Extracts from this document...

Introduction

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Shape 1

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Shape 2        

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Shape 3        

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image04.pngimage04.pngimage04.pngimage06.pngimage05.pngimage04.pngimage02.png

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Shape 4image04.pngimage04.pngimage04.pngimage04.png

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A shape, that I have named Shape 3 on the previous page, was given to me and my task was to find the correlation between the shape number and the amount of squares needed to make this shape.

At first glance I named the shape, Shape 1, but after studying the pattern I realised that I could draw two smaller shapes that still followed the way in which the original shape was created.

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image13.pngimage13.pngimage12.pngimage12.png

         Row            No. of cubes     Total

Shape 1:   1                  1  

                                   2                  3                          5        

  1. 1                                            

Shape 2:  1                   1

  2                   3

  3                   5                          13

           4                   3

            5                   1image14.png

Shape 3:                    1                   1

            2                   3

            3                   5

            4                   7                           25

            5                   5

            6                   3

            7                   1image14.png

Shape 4:                    1                   1

            2                   3

            3                   5

            4                   7

            5                   9                 41

            6                   7

            7                   5

            8                   3

            9                   1

What I have noticed was that instead of writing down each number of squares in each

row to find the total I could use a special method:

                        2[sum of n consecutive odd no’s] + 2n + 1

For example:

Shape no. 2 ~  2[ 1 +  3 ] + 2 x 2 + 1

                =     2  +  6   +    4    + 1

                =   13

image15.png

I am going to start by trying to find the nth term by using the method below:

        No of total squares:        5        13        25        41image16.png

1st diferrences not the same:             8        12     16

2nd differences arethe same:                 4         4

If the 2nd difference is a constant, the formula for the n

...read more.

Middle

Rest of Sequence            3             5                   7              9

   2                       2                  2

This difference of 2 is the number which goes infront of n.  This forms the second part of the formula.

To find the third and last part of the formula, which will be a number on it’s own, I will use the 1st and 2nd part of the formula that I have already obtained:

nth term = 2n² + 2n + ?

1st term ~ 2x1² + (2x1) + ? = 5

                                               2     +    2    + ? = 5

                                                                 ? = 1

                                 2nd term ~ 2x2² + (2x2) + ? = 13

                                               8     +   4     + ? = 13

                                                                 ? = 1

                                3rd term ~ 2x3² + (2x3) + ? = 25

                                                      18    +   6     + ? = 25

                                                                        ? = 1

This number is a constant therefore the number 1 is the last part in the formula.

image20.png

 The formula for the nth term is:                  2n² + 2n + 1image21.png

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When I thought about the total number of squares, I realised that this, in other words, was the same as saying the number of black squares + the number of white squares.

This gave me the idea for doing the following:

I am going to find the equation for the number of white squares:

image23.png

                Shape Number           1           2                  3            4            n

           No of white squares           4              8                 12          16        4n

 the equation for the number of white squares is:    4nimage24.png

...read more.

Conclusion

                2                                                        2

144        OR-146

              2                                                          2

image24.pngimage24.png

                                 72     OR         -73

A: Yes, 10513 does make up a shape number.  The shape number is 72.  It      

     can’t be  - 73 as it would be impossible to have a minus value shape  

     number.

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On receiving this task, I thought it would make this exercise easier to understand if I built my own 3D model by using multi-link cubes.  By this way I was able to take off each layer and count the cubes that lay underneath.  I thought this was a very practical and helpful way of dealing with this particular task.

While I was doing this I noticed that in each layer, instead of writing the total as a number, I could substitute it with two squared, consecutive numbers, similar to what I have done in part 1.

These are my results of the different amount of cubes in each layer of each shape number that I have focused on in part 1:

image36.png

                    Layer           No. of cubes       Squared No’s        Total

Shape 1            1                           1                    0² + 1²                        

  1.  5                    1² + 2²                   7
  2.  1                    0² + 1²

Shape 2            1                           1                    0² + 1²

    2                            5                   1² + 2²

    3                          13                   2² + 3²                  25

    4                             5                   1² + 2²

    5                           1                    0² + 1²

Shape 3  1                           1                    0² + 1²                                                                                

    2                           5                    1² + 2²                                                      

    3                         13                    2² + 3²                                                      

    4                         25                    3² + 4²                  41                              

    5                         13                    2² + 3²                                              

    6                           5                    1² + 2²                                                

    7                           1                    0² + 1²                                                  

Shape 4           1                          1                    0² + 1²  

   2                           5                    1² + 2²

   3                         13                    2² + 3²

   4                         25                    3² + 4²

   5                         41                    4² + 5²                 129

   6                         25                    3² + 4²                                                

   7                         13                    2² + 3²                                                                      

   8                           5                    1² + 2²                                                    

   9                           1                    0² + 1²                                  

What I noticed was:

In Shape 1 there were – three 1² = 3 (1²)      

                                  one  2² = 2²

In Shape 2 there were -   four 1² = 4 (1²)

                               three 2²  = 3 (2²)

                                        one 3² =  3²                      

In Shape 3 there were -  four 1² = 4 (1²)

                                   four 2² = 4 (2²)

                                three 3² = 3 (3²)

                                         one 4² = 4²

In Shape 4 there were -   four 1² = 4 (1²)

                                   four 2² =  4 (2²)    

                                 four 3² =  4 (3²)      

                                  three 4² =  3 (4²)      

                                    one 5² = 5²                                                                                                                                                                                                          

...read more.

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