Quadratic patterns

e.g.  Find a rule for calculating the total number of boxes
        in each diagram, if the number of boxes in the
        bottom layer is known. 



        Draw up a table to show the number of boxes in the
        bottom layer and the total number of boxes in each
        diagram.

        Calculate the differences between the terms.

No. of boxes in bottom layer (n)	1	2	3	4	5
Total no. of boxes (t)	1	3	6	10
First difference	2    3    4
Second difference	1    1

Let the number of boxes in the bottom layer be n and the total number of boxes be t.

Because the second differences are the constant (they are all equal to 1), the rule in this example will be a quadratic expression.

The rule will be in the form: t = an² + bn + c – where a, b, and c are constants to be found.

Now when n=1, t=1 so 1 = a + b + c -------- (1)

When n=2, t=3 so 3 = 4a + 2b + c -------- (2)

When n=3, t=6 so 6 = 9a + 3b + c -------- (3)

Subtract equation (1) from equation (2) to eliminate c:
2 = 3a + b -------- (4)

Subtract equation (2) from equation (3) to eliminate c:
3 = 5a + b -------- (5)

This now gives two equations in a and b.
Solve these simultaneously to find a and b.

Subtract equation (4) from equation (5):
1 = 2a
a = (1/2)
Substitute this value for a in equation (4) to find b = (1/2)

Substitute these values for a and b into equation (1) to find c = 0.

Write a = (1/2), b = (1/2) and c = 0 into the rule.
Hence the rule for this pattern is t = (1/2)^n² + (1/2)ⁿ