# 2D and 3D Sequences Project.

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:

=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

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.

Attempting to Obtain a Formula Through the Use of the Difference Method

I will now apply Jean Holderness' difference ...