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

My job is to investigate how many squares would be needed to make any cross shape like this build up in the same way.

Extracts from this document...

Introduction

Adeel Younis, Math’s Coursework Borders, Mr. Ekwalla, 10A, 10G 5/10/2007

Borders: Part 1

image05.png

My job is to investigate how many squares would be needed to make any cross shape like this build up in the same way.

Below are diagrams of the cross shape pattern to the 8th sequence:

image06.png

 Here is my table of results, with the number of black, red and total squares

Black Squares

4

8

12

16

20

24

28

32

Red Squares

1

5

13

25

41

61

85

113

Total Squares

5

13

25

41

61

85

113

145

Now so I can find an equation which will tell me how many squares there will be in each sequence, I will find the differences for the black squares first.

image00.png

To find the equation I will need bits of information, the equation Is an+b, where a is the difference and b is the 0th term, e.g. the first term here is 4, to find the 0th

...read more.

Middle

st step find the differences

2nd Step find the equation, a=2 c=1 now to find b

1)Un=an²+bn+c

2)5= an²+bn+c

3)5=2×1²+b×1+1

4)5=2+b+1

5)5=3+b

6)5-3=3+b-3

7)2=b

Let’s test it; term 1 is 5, 2×1²+2×1+1=5. I’m right again! Now we have the equations for the black, white and total squares

Black Squares= 4n+0

Red Squares = n=2n²+-2n+1

Total Squares = n=2n²+2n+1

Borders: Part 2

Now to expand my investigation into the 3rd dimension. Please find the sheet with my drawing of the 3D shapes on, after you have seen that continue reading

------------------------------------------------------------------------------------------------------------

Here is where maths becomes complicated, the drawings on the last page are the cross shaped in the 3rd dimension, and here is the table showing how many cubes each shape has.

Shape

1

2

3

4

5

6

No. of cubes

7

25

63

129

231

377

 Now I will find the differencesimage02.png

...read more.

Conclusion

tml/images/image10.png" style="width:16px;height:41.33px;margin-left:0px;margin-top:0px;" alt="image10.png" />

36

85 image15.png

116 image10.png

288

S-4/3n³

5 image07.png

14 image08.png

27

43 image07.png

64 image08.png

89

 Here we go now were left with the quadratic equation, now I will find its differences.

Now we can find b, c and d. b, c and d are basically the a, b and c in the quadratic equation, in the same way we have to find c last and b and c first, b=½ of the second difference, in this case 2, d= 0th term, in this case 1, now we substitute b and d into the equation to find c:

1)Un=bn²+cn+dimage03.png

2)27= an²+cn+c

3)27=2×3²+3c×3+1

4)27=18+3c+1

5)27=19+3c

6)27-19=19+3b-19image04.png

7)8=3b

8)image09.png

9)2image10.png=c

Now we have c, the full equation is image11.pngn³+2n²+image12.pngn+1. Our journey doesn’t end here, we still need to test it, let’s go ahead. We will test it with the 1st term, the 1st term is 7, image11.png×1³+2×1²+image12.png×1+1=7 Hooray, the equation works.

 of

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics 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 AS and A Level Core & Pure Mathematics essays

  1. 2D and 3D Sequences Project Plan of Investigation

    128 - 15 5. = 113 Successful The formula I found seems to be successful as I have shown on the previous page. I will now use the formula to find the number of squares in a higher sequence. So now I wil use the formula 2n2 - 2n + 1 to try and find the number of squares contained in sequence 20.

  2. Numerical Method (Maths Investigation)

    For DECIMAL SEARCH X F(X) 1.51211 =EXP(X_)-3*X_ 1.51212 =EXP(X_)-3*X_ 1.51213 =EXP(X_)-3*X_ 1.51214 =EXP(X_)-3*X_ 1.51215 =EXP(X_)-3*X_ 1.51216 =EXP(X_)-3*X_ 1.51217 =EXP(X_)-3*X_ 1.51218 =EXP(X_)-3*X_ 1.51219 =EXP(X_)-3*X_ Tbl AppA-01: Formula used in Microsoft Excel XP for decimal search From the table above, under the f(X)-value column, it all contains the formula =EXP(X_)-3*X_.

  1. Methods of Advanced Mathematics (C3) Coursework.

    On the first route this idea becomes clear as we can see by the highlighted red cells that for the first value of the root drawn out by the equation becomes the starting point for the next set of figures to go through the iterative equation Route 1 xn f(xn)

  2. Triminoes Investigation

    27a + 9b + 3c + d = 20 - equation 3 - 8a + 4b + 2c + d = 10 - equation 2 19a + 5b + c = 10 - equation 6 Equation 2 - Equation 1 I am doing this again to eliminate d from the two equations, to create another equations.

  1. Functions Coursework - A2 Maths

    To illustrate the fact that the root lies in the interval [1.87,1.88], part of the graph of y=f(x) is drawn. The graph crosses the x-axis visibly between x=1.87 and x=1.88. This means that between x=1 and x=2, y=f(x)=0 for a value of x in the interval [1.87,1.88] i.e.

  2. Arctic Research (Maths Coursework)

    this position taking into consideration the westerly wind, and allow me alter the position of the base camp afterwards in order to find the best location within the circle where the journey times to all 8 of the observation sites are limited.

  1. Sars Math Portfolio ...

    - exponential function ( y = abx) - logarithmic function ( y = n logxy) The first type of function that will be used is the exponential function. y = abx We will have to solve for a and b using simultaneous equations. We start by inserting figures for day 28 and day 0.

  2. GCSE Math Coursework: Triminoes

    Using number 0 (000) Using numbers 0 and 1 (000) (001) (011) (111) Using numbers 0, 1 and 2 (000) (001) (002) (011) (012) (022) (111) (112) (122) (222) Using numbers 0, 1, 2 and 3 (000) (001) (002) (003) (011) (012) (013) (022) (023) (033) (111) (112) (113) (122)

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