So my formula now is: Un = n*+bn+c, to find b I have to make a new sequence.
Term 1 2 3 4 5
Sequence 1 3 7 13 21
n* 1 4 9 16 25
New 0 -1 -2 -3 -4
Sequence
1st difference –1 -1 -1 -1 -1
And B Is the difference of the new sequence so b= -1 our formula now is
Un = n*- n + c
Now I have to find c we will have to put the formula into action:
If n = 1
U1 = 1* – 1 + c and because u = 1 c must be +1 so the formula must be
Un = n* - n + 1 My predicted formulae is n = n* – n + 1
This is the formula for the maximum number of regions
Here is a table of results to show if the formula works, U is the term.
I will test my formula to find out the maximum number of regions for 5 circles
U5 = 5* - 5 + 1
U5 = 5 x 5 – 5 + 1
U5 = 25 – 5 + 1 = 21
I can see that this answer follows the pattern
However I will still draw 5 circles to test my prediction
The maximum number of regions
For 5 circles is 61
I will now try to do the same with triangles
The Maximum Number When 3 triangles When 4 triangles
Of regions created when overlap a maximum overlap a maximum
2 triangles overlap is 7 of 19 can be made. Of 37 can be made
Already I can see that there is a pattern developing that is similar to that of the circles, however the old formula does not work:
U2 = 2* - 2 + 1 = 2
so it is not the same as the circles however I can work out what the formula is by using the same methods:
Term U 1 2 3 4
Sequence 1 7 19 37
1st Difference 6 12 18
2nd Difference 6 6
Now from looking at this set of results I can see that this formula will be quadratic. Now to find my new formula I must do as I did first time round and half my 2nd difference, which means it is 3 so, my formula so far is now.
Un = 3n – bn + c
Now using the same method as before I will find out what b is
Term 1 2 3 4
Sequence 1 7 19 37
3n* 3 12 27 48
New -2 -5 -8 11
Sequence
1st -3 -3 -3
Difference
I can now see that b is –3 so my formula is now:
Un = 3n – 3n + c
Now if I follow the same method as before c = 1 so my final formula is:
Un = 3n – 3n + 1
I will now try to find the formula for the maximum number of regions for squares.
The maximum number the maximum number the maximum number
Of regions created when when 3 overlap is 25 when 4 overlap is 49
2 squares overlap is 9
I can see once again that the same pattern as the circles is developing so I know that the circle formula does not work and I can tell that the triangle formula does not work, as the numbers are too different. So I will have to find a new formula.
Term 1 2 3 4
Sequence 1 9 25 49
1st difference 8 16 24
2nd difference 8 8
Once again I can see that my formula must be quadratic and once again I must half the 2nd difference and so my new formula for squares is
Un = 4n – bn + c
Now I must do the same as I did in the last two to find out b.
Term 1 2 3 4
Sequence 1 9 25 49
4n* 4 16 36 64
New -3 -7 -11 -15
Sequence
2nd
Difference -4 -4 -4
I can now see that –4 is b so my new formula is Un = 4n – 4n + c
And I can also see that c will be +1 so my formula is now
Un = 4n – 4n + 1
This is my final formula for squares I will now test it for 5 squares
U5 = 4 x 5* - 4 x 5 +1
U5 = 100 – 15 + 1 = 81
This is correct as it follows the pattern so that is my final formula for squares. I do not need to draw 5 squares as I know this is right as it worked for the circles
Conclusion
After looking over and analysing my results I can see that a & b is always the same and is always equal to the amount of sides the shape has for example triangle 3n and square 4n the circles did not show this as circles have no sides I have also found the general formula for finding out how many regions are in any shape is:
Un = an*+ bn+c
