Carl formed some figures using squares and circles as shown.

1. Find the number of squares in Figure 8.
2. Find the total number of circles and squares in Figure 13.
3. A figure in the pattern has a total of 2071 circles and squares. What is the Figure Number?

(a)
Number of squares
Figure 1: 2 = 1 + 1
Figure 2: 3 = 1 + 2
Figure 3: 4 = 1 + 3
Figure 4: 5 = 1 + 4

Formula:
Number of squares = 1 + Figure Number

Number of squares in Figure 8
= 1 + 8
= 9

(b) 
Total number of circles and squares
Figure 1: 3   = 1 + 2(1)
Figure 2: 7   = 1 + 2(1 + 2)
Figure 3: 13 = 1 + 2(1 + 2 + 3)
Figure 4: 21 = 1 + 2(1 + 2 + 3 + 4)

Formula:
Number of circles and squares= 1 + 2[¹₂ x Figure Number x (Figure Number + 1)]

Total number of circles and squares
= 1 + 2[¹₂ x Figure Number x (Figure Number + 1)]
= 1 + 2(¹₂ x 13 x 14)
= 1 + 13 x 14
= 1 + 182
= 183

(c)
1 + 2[¹₂ x Figure Number x (Figure Number + 1)] = 2071
2[¹₂ x Figure Number x (Figure Number + 1)] = 2071 - 1
Figure Number x (Figure Number + 1) = 2070
45 x 46 = 2070

Figure number that has a total number of circles and squares = 45

Answer(s): (a) 9; (b) 183; (c) 45