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Sequences and series investigation
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Introduction
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 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)
1(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.
Middle
3. 18 -5
4. = 13
The formula when applied to sequence 3 appears to be
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.
Using the Difference Method to Find an Equation to Establish
the Number of Squares in a 3D Version of the Pattern
Pos.in seq. 0 1 2 3 4 5
No.of squar. -1 1 7 25 63 129
1st differ. 2 6 18 38 66
2nd differ. 4 12 20 28 36
3rd differ. 8 8 8 8
So therefore we get the equation;
anƒ + bn2 + cn + d
We already know the values of 'n' (position in sequence) in the
equation so now we have to find out the values of a, b, c, and d.
If n = 0 then d = -1 and if n = 1 then d = 1
I can now get rid of d from the equation to make it easier to find
Conclusion
was a quadratic. This can be proven through the fact that the 2nd
difference was a constant, a necessary element of any quadratic
and also the fact that the first value has to be squared. This can
also be proved by illustrating the equation on the graph, creating a curve.
I have also established that the top triangular half of the 2D pattern
always turns out to be a square number.
If we now look at the 3D pattern, the equation I achieved for it
has turned out to a cubic equation. This can be proven through the
constant, again a necessary characteristic of any cubic equation
and also the fact that its 1st value must be cubed and its second
squared. If we drew a graph we would get a ccurved graph in which
the line falls steeply, levels off and then falls again.
The Differentiation Method developed by Jean Holderness played a
very important role in this investigation. It helped us to gain
knowledge of any pattern and anything that would help in the invetigation,
giving us our constant, but most importantly it gave us the equation
on which to base our solutions.
It was:
an2 + bn + c
This proved very helpful.
To find our equation we then substituted in different values which
we could find in our differentiation table.
I have concluded that both the equations proved to be very successful.
Therefore the equations are:
For the 2D pattern the equation is;
2n2 - 2n + 1
For the 3D pattern the equation is;
_(nƒ) - 2n2 + 2Y(n) - 1
This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.
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