2A = 4 A = 2 Nth term = 2N2 - 6N + 5
3A + B = 0
6 + (-6) = 0
A + B + c = 1
2 + (-6) + 5 = 1
To make sure that I have worked out all my equations properly I am now going to draw a test diagram to make sure. I predict that the 6th diagram in the sequence will have.
N = 6 because it is the 6th diagram, if it was the 7th diagram then N = 7 and so on.
I am now going to duplicate my results for another shape (triangles) to see If the sequences are the same, if not I will look for a pattern between them, a general term
Triangles
I will now put all the figures I have recorded into tables and find the Nth term
I have noticed that on the 2nd difference the numbers are repeating (3, -3, 3, -3….) because of this there is no way to be sure that further on down in the differences that it will ever be the same. I have found that by using 2 Nth terms 9(N/2 + 0.5) - 6 and 9(N/2) - 3
For the sequences in triangles I will use simultaneous equation to work out the Nth term.
A + B + c = 1 ---------(1)
4A + 2B + C = 4 ---------(2)
9A + 3B + C = 10 --------(3)
To work this equation out I will have to eliminate one of the letters. I will remove the +c because it is in all the equations
((4) X 2) - (5)
To find B I will put A back into equation (4)
To find C I am now going to put A and B into equation (2)
I will now test all three in equation (1)
This is wrong, as I can't see where I have gone wrong I will use the same method I used in the sequences for the blocks. I have decided to use the same method, as before because it is the one for which I am most familiar therefore I am less likely to go wrong. I think that by doing it out as a simultaneous equation it is inefficient and takes too long with a larger margin for error than other methods.
2A = 3 A = 1.5
3A + B = 3
(1.5 X 3) + B = 3
4.5 + (-1.5) = 3 B = -1.5
A + B + C =1
1.5 + (-1.5) + 1 C = 1
So the equation is 1.5N2 - 1.5N + 1
So the equation is 3N - 3
2A= 3 A= 1.5
The Nth term for unshaded is 1.5N2 -4.5N + 4
I am now going to test my sequences against the 6th diagram in the triangle sequence. I predict that it will have
For diagram see triangular spotty paper
General rule
Now that I have worked out all the Nth terms for both blocks and triangles I am going to try and find a general rule for the Nth in each sequence.
Perimeter
Blocks = 8N - 4
Triangles = 9N + 6 and 9N - 3
Area
Blocks = 2N2 - 2N + 1
Triangles = 1.5N2 - 1.5N + 1
Shaded
Blocks = 4N - 4
Triangles = 3N - 3
Unshaded
Blocks = 2N2 - 6N + 5
Triangles = 1.5N2 - 4.5N + 4
I have looked at these equations and noticed that they can be broken down by dividing the equation by the number of sides in a shape. I can't work this out for the perimeter because of the equation I had to use for the triangles. I think that if I use my other equations I can find the sequences for another shape without drawing it out. I have chosen to try and work the equations out for a hexagon. But first I will write out the general terms.
Use these equations to work out the following (where x is the number of sides)
Area - X (0.5N2 - 0.5N + (1/X)) i.e. for a block this would be
2N2 - 2N + 1
Shaded - X (N - 1) i.e. for a triangle this would be
3N - 3
Unshaded - X (0.5N2 - 0.5N + (1 1/X)) i.e. for a block this would be
2N2 - 6N + 5
I predict using my general terms that a hexagon would have the sequences
Area - 3N2 - 3N +1
Shaded - 6N - 6
Unshaded - 3N2 - 9N + 7
Using my prediction of general terms I have worked out the perimeter for the 5th diagram. To make sure that it is correct I will draw out diagrams 1 - 5 and work out the sequences using my predicted equations.
5th Diagram
Area - 3N2 - 3N + 1
(3 X 52) - (3 X 5) + 1
(3 X 25) - (3 X 5) + 1
75 - 15 + 1 = 61
Shaded - 6N - 6
(5 X 6) - 6
30 - 6 = 24
Unshaded - 3N2 - 9N + 7
(3 X 52) - (9 X 5) + 7
(3 X 25) - (9 X 5) + 7
75 - 45 + 7 = 37
Perimeter - I was unable to find a general term for the perimeter so I will work it out
General term for the perimeter is 12N - 6
Conclusion
During this investigation I have looked at different ways of solving numeric sequences derived from patterns found in shapes. From the sequence I was originally given I have worked out Nth terms that help describe the patterns in the shape (Perimeter, Area, Number of shaded and the Number of unshaded). I moved on to another shape (triangles) to see if the pattern continued similarly in other tessellating shapes. I found that the patterns might not have been the same, but by dividing the sequences by the number of sides in each category, I could get a general Nth term. This general Nth term can be used to find the pattern in any shape simply by multiplying the general term by the number of sides in the shape you are investigating. I looked to see if there was a single term that could be used to describe the pattern in all shapes but I was unable to find it.
