Investigating Shape Patterns
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My job is to investigate how many squares would be needed to make any
cross shape like this build up in the same way.
Below are diagrams of the cross shape pattern to the 8th sequence:
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Here is my table of results, with the number of black, red and total
squares
Black Squares
4
8
12
16
20
24
28
32
Red Squares
1
5
13
25
41
61
85
113
Total Squares
5
13
25
41
61
85
113
145
Now so I can find an equation which will tell me how many squares
there will be in each sequence, I will find the differences for the
black squares first.
The 1st difference is constant; I can now make an equation
To find the equation I will need bits of information, the equation Is
an+b, where a is the difference and b is the 0th term, e.g. the first
term here is 4, to find the 0th terms you need to go back one step
(take the difference from the first terms to find the 0th term) and n
is the term you want to find, e.g. if you want to find the 1st term
n=1 if you want to find the second term n=2.
a=4, b=0. Now I have the equation, 4n+0, now let's test it! We know
the first terms is 4, so 4xn(1)+0, 4x1=4 so the answer is 4 and that
is the 1st term.
Now ill do the same with the red squares
Hey, this time the 1st difference isn't constant, ill need to find the
2nd difference and a different equation
To find this equation I will need different bits of information, the
equation is an2+bn+c,
A is ½ of the second difference, in this case a = 2
=> rd£ = (1 + rd$)*(1 + i£)/(1 + i$) -1 = (1+ rd$)*1.043/1.027 -1
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