# 2D & 3D Sequences.

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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.

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.

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(62) - 12 + 1

2. 2(36) - 12 +1

3. 72 - 12 + 1

4. 72 - 11

5. = 61

Successful

Sequence 8:

1. 2(82) - 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 2n2 - 2n + 1 to try and find

the number of squares contained in sequence 20.

Sequence 20:

2 (202) - 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.

This student written piece of work is one of many that can be found in our GCSE Consecutive Numbers section.

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