Ivan formed some figures using squares and circles as shown.
- Find the number of squares in Figure 12.
- Find the total number of circles and squares in Figure 22.
- A figure in the pattern has a total of 3081 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 12
= 1 + 12
= 13
(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 22 x 23)
= 1 + 22 x 23
= 1 + 506
= 507
(c)
1 + 2[
12 x Figure Number x (Figure Number + 1)] = 3081
2[
12 x Figure Number x (Figure Number + 1)] = 3081 - 1
Figure Number x (Figure Number + 1) = 3080
55 x 56 = 3080
Figure number that has a total number of circles and squares = 55
Answer(s): (a) 13; (b) 507; (c) 55