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  • Level: GCSE
  • Subject: Maths
  • Word count: 1849

3 step stair investigation

Extracts from this document...

Introduction

Mathematics Coursework

3 Step Stair Investigation

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N+20

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N+11

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N+1

N+2

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N+18

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N+10

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N+1

N+2

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N+16

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N+2

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N+14

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N+12

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I can already observe a pattern therefore I should be able to write a final formula for a 3 by 3 step in any size grid so far the answers I have are:

image06.png

10 by 10→ 6N + 44

9   by 9  → 6N + 40

8   by 8  → 6N + 36

7   by 7  → 6N + 32

6   by 6  → 6N + 28

image07.pngimage08.png

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N+30

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N+27

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N+3

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N+24

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image16.pngimage14.png

N+21

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There is an obvious relationship between the answers I have got for different size grids therefore I should now attempt to write a formula for a 4 by 4 step in any size grid.

image17.png

10 by 10

...read more.

Middle

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N+40

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N+36

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There is an obvious relationship between the answers I have got again for different size grids therefore I should now attempt to write a formula for a 5 by 5 step in any size grid.image33.png

10 by 10 15N + 220

9   by 9   15N + 200

8   by 8   15N + 180

7   by 7   15N + 160

6   by 6   15N +?

? Should equal 140. To prove my formula I will check it works in a 6 by 6 grid.

image34.png

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...read more.

Conclusion


(∑Tn < X) (y+1)

 This is the final formula for the second part of the investigation I already have the first so if I were to write out the complete formula it would be.

TnX+ (∑Tn < TnX)(y+1)  N = the of the shape and X = the corner number                                                       Y= size of the grid  (But there is another way)

image75.png

By using the cubic formula I will be able to calculate the relationship between these numbers in red.

1 step stair = 1N   + 0(y+1)

2 step stair = 3N   + 1(y+1)

3 step stair = 6N   + 4(y+1)

4 step stair = 10N + 10 (y+1)

5 step stair = 15N + 20(y+1)

6 step stair = 21N + 35(y+1)

an3 +bn2 +cn+ d

0                        1                                4                                10

             1                                3                                6

                        2                                3

                                        1

a+b+c+d       8a+4b+2c+d              27a+9b+3c+d            64a+16b+4c+d

        0                     1                                4                                  10

           7a+3b+c                 19a+5b+c                   37a+7b+c

                    1                            3                                 6

                            12a+2b                      18a+2b

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                 1

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a= 1/6

b= 0image79.png

c= -1/6

d= 0

image80.pngimage80.png

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TnX +                                   (y+1)

I now have a final formula which hopefully works all I now need to do is test it. To begin with I will imagine I have a 3 by 3 step stair in a 10 by 10 grid. I know the answer for the formula should be 6x +44.

T3X= 6x

image82.png

                                             24

                        =     ----- =   4

                                              6

6x+ 4 (10+1) = 6x + 44

If the corner number is one then the answer would be 50 which is the correct answer.

1+2+3+11+12+21=50

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image05.pngimage04.pngimage88.pngimage89.pngimage90.pngimage91.pngimage92.pngimage93.pngimage03.png

Roy Vivasi                          Mr Clear                      

...read more.

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