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  • Level: GCSE
  • Subject: Maths
  • Word count: 1497

Borders - find out the differences in the patterns of the colored squares.

Extracts from this document...

Introduction

George Eghator 11

...read more.

Middle

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Table of Results

Num.          num black                   num white                             total

1.        0        1        1

2.        1        4        5

3.        5        8        13

4.        13        12        25

5.        25        16        41

6.        41        20        61

7.        61        24        85

8.        85        28        113

9.        113        32        145

N                      Total                                       Structure

1        1        (1+1)2+1²

2        5        (2+1)2+2²

3        13        (3+1)2+3²

4        25        (4+1)2+4²

5        41        (5+1)2+5²

6        61        (6+1)2+6²

7        85        (7+1)2+7²

8        113        (8+1)2+8²

9        145        (9+1)2+9²

Order of the squares

1 + 3 + 1 = 5

1 + 3 + 5 + 3 + 1 = 13

In each of these cases, 2 have been added.

1 + 3 +5 7 + 5 + 3 + 1 = 25

As seen in the other pattern, there are additions with 2 being added on.

For the next pattern, I think that the total number of squares will be 41, using the following pattern:

1 + 3 + 5 + 7 + 9 + 7 + 5 + 3 + 1 = 41

Now I am going to test my prediction.

Number of squares = 25 + 16 = 41

Pattern

Dark squares

White squares

1

1

4

1 + 4 = 5

2

5

8

5 + 8 = 13

3

13

12

13 + 12 = 25

4

25

16

25 + 16 = 41

This result is made because we are just adding onto it.

Differences

Total

5     13      25      41     61      85      113     145   181    221

1st difference

   8      12      16      20      24      28      32      36      40

2nd difference

      4         4      4         4         4         4        4       4             

From the quadratic sequence, we see that the main difference is 4. The first formula I will try to find is the formula for the surrounding white squares.

...read more.

Conclusion

1. There will be an nth term for the number of white squares

2. There will be an nth term for the number of black squares

3. There will also be an nth term for the amount of colored squares

These are my theories which will be shown throughout the investigation as evidence.

C.  Plan of action

My plan of action will be to give indication to the methods and formulas that I will use. Algebraic expressions such as nth term will be used because it will be reliable towards my investigation.

Here are some of the expressions that will be used:

Quadratic formula- an2 + bn + c

Cubic formula- an3+ bn2+ cn + d

Using 3D formulas to find invisible patterns.

I will predict the variable sequences and compare results to analyze and explain them. Formulas for the patterns will be involved such as cubic and quadratic. Having this will help me to understand the concepts of my geometric investigation.  

        -  -

...read more.

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