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2D & 3D Sequences.

Extracts from this document...

Introduction

2D & 3D Sequences

Plan of Investigation


In this experiment I am going to require the following:

A calculator
A pencil
A pen
Variety of sources of information
Paper
Ruler

In this investigation I have been asked to find out how many squares
would be needed to make up a certain pattern according to its sequence.
The pattern is shown on the front page. In this investigation I
hope to find a formula which could be used to find out the number
of squares needed to build the pattern at any sequencial position.
Firstly I will break the problem down into simple steps to begin
with and go into more detail to explain my solutions. I will illustrate
fully any methods I should use and explain how I applied them to
this certain problem.

...read more.

Middle


have seen a number of different patterns.

Firstly I can see that the number of squares in each pattern is an odd number.

Secondly I can see that the number of squares in the pattern can
be found out by taking the odd numbers from 1 onwards and adding
them up (according to the sequence). We then take the summation
(å) of these odd numbers and multiply them by two. After doing this
we add on the next consecutive odd number to the doubled total.

I have also noticied something through the drawings I have made
of the patterns. If we look at the symetrical sides of the pattern
and add up the number of squares we achieve a square number.



...read more.

Conclusion


successful.

Sequence 5:
1. 2(52) - 10 + 1
2. 2(25) - 10 + 1
3. 50 - 10 + 1
4. 50 - 9
5. = 41
Successful

Sequence 6:
1. 2(6
2) - 12 + 1
2. 2(36) - 12 +1
3. 72 - 12 + 1
4. 72 - 11
5. = 61
Successful

Sequence 8:
1. 2(8
2) - 16 + 1
2. 2(64) - 16 + 1
3. 128 - 16 + 1
4. 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 2n
2 - 2n + 1 to try and find
the number of squares contained in sequence 20.

Sequence 20:

2 (20
2) - 40 + 1
2(400) - 40 + 1
800 - 40 + 1
800 - 49
= 761

Instead of illustrating the pattern I am going to use the method
I used at the start of this piece of coursework. The method in which
Iused to look for any patterns in the sequences. I will use this
to prove the number of squares given by the equation is correct.
As shown below:

2(1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35+37) + 39 = 761

I feel this proves the equation fully.














...read more.

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