The first four figures of a pattern as shown.

The table shows the number of circles and squares used in each figure.

1. Find the values of (i) and (ii). Give your answers in the following format. (Eg 1, 2)
2. How many circles are there in Figure 30?
3. How many squares are there in Figure 69?

(a)
(i)
Number of circles:
Figure 1:  5  = 2 + 1 x 3
Figure 2:  8  = 2 + 2 x 3
Figure 3: 11 = 2 + 3 x 3

Formula:
Number of circles = 2 + Figure Number x 3

Number of circles for Figure 5
= 2 + 5 x 3
= 2 + 15
= 17

(ii)
Number of squares:
Figure 1: 4   = 4 x 1
Figure 2: 12 = 4 x (1 + 2)
Figure 3: 24 = 4 x (1 + 2 + 3)

Formula:
Number of squares = 4 x ^{Figure Number x (Figure Number + 1)} ₂

Number of squares at Figure 5
= 4 x ^{Figure Number x (Figure Number + 1)} ₂
= 4 x ^{5 x (5 + 1)} ₂
= 4 x ^{5 x 6} ₂
= 2 x 30
= 60

(b)
Number of circles in Figure 30
= 2 + 30 x 3
= 2 + 90
= 92

(c)
Number of squares in Figure 69
= 4 x ^{Figure Number x (Figure Number + 1)} ₂
= 4 x ^{69 x (69 + 1)} ₂
= 4 x ^{69 x 70} ₂
= 2 x 4830
= 9660

Answer(s): (a) 17, 60; (b) 92; (c) 9660