Sequences and series investigation By Neil
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 sequential 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
them up (according to the sequence). We then take the summation
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 sequential 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
them up (according to the sequence). We then take the summation