Growing Squares
I have decided to find a formula to find the nth term. To help me find
the nth term I shall compose a table including all the results I know.
Pattern Number
Number of Squares
1st Difference
2nd Difference
1
1
4
4
2
5
8
3
13
4
12
4
25
The 2nd difference is constant; therefore the equations will be
quadratic. The general formula for a quadratic equation is an2 + bn
+c. The coefficient of n2 is half that of the second difference
Therefore so far my formula is: 2n2 + [extra bit]
I will now attempt to find the extra bit for this formula.
Pattern Number
Extra Bit
1st Difference
2nd Difference
1
2
6
4
2
8
10
3
18
4
14
4
32
From my table of results I have found the formula to be 2n2 + 2n + 1
I will now check my formula by substituting a value from the table in
to my formula:
E.g. n = 2
Un = 2 (2) 2 - 2 (2) + 1 = 8.
For Diagrams 1 - 4 I can see a pattern with square numbers. The
diagrams numbers squared added to one less than the diagram number
squared gives the correct number of squares.
For diagram n it should be:
Un = (n - 1) 2 + n2
(n - 1)(n - 1) + n2
n2 - n - n - n + 1 + n
Un = 2n2 - 2n + 1
This is correct.
Growing Hexagons
I will now repeat my investigation, and change the original shape of
the square to hexagons, and try to find the formula as before.
I shall start by finding the
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So now that we have the desired answer and know what the formula is, we just need to know what it means and why it works. So
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