2D and 3D Sequences Project

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. I will firstly carry out this experiment on

a 2D pattern and then extend my investigation to 3D.

The Number of Squares in Each Sequence

I have achieved the following information by drawing out the pattern and extending upon it.

Seq. no. 1 2 3 4 5 6 7 8

No. Of cubes 1 5 13 25 41 61 85 113

I am going to use this next method to see if I can work out some sort of pattern:

Sequence Calculations Answer

=1 1

2 2(1)+3 5

3 2(1+3)+5 13

4 2(1+3+5)+7 25

5 2(1+3+5+7)+9 41

6 2(1+3+5+7+9)+11 61

7 2(1+3+5+7+9+11)+13 85

8 2(1+3+5+7+9+11+13)+15 113

9 2(1+3+5+7+9+11+13+15) +17 145

What I am doing above is shown with the aid of a diagram below;

If we take sequence 3:

2(1+3)+5=13

2(1 squares)

2(3 squares)

(5 squares)

The Patterns I Have Noticied in Carrying Out the Previous Method

I have now carried out ny first investigation into the pattern and

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

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. I will firstly carry out this experiment on

a 2D pattern and then extend my investigation to 3D.

The Number of Squares in Each Sequence

I have achieved the following information by drawing out the pattern and extending upon it.

Seq. no. 1 2 3 4 5 6 7 8

No. Of cubes 1 5 13 25 41 61 85 113

I am going to use this next method to see if I can work out some sort of pattern:

Sequence Calculations Answer

=1 1

2 2(1)+3 5

3 2(1+3)+5 13

4 2(1+3+5)+7 25

5 2(1+3+5+7)+9 41

6 2(1+3+5+7+9)+11 61

7 2(1+3+5+7+9+11)+13 85

8 2(1+3+5+7+9+11+13)+15 113

9 2(1+3+5+7+9+11+13+15) +17 145

What I am doing above is shown with the aid of a diagram below;

If we take sequence 3:

2(1+3)+5=13

2(1 squares)

2(3 squares)

(5 squares)

The Patterns I Have Noticied in Carrying Out the Previous Method

I have now carried out ny first investigation into the pattern and

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