Billy formed some figures using squares and circles as shown.
- Find the number of squares in Figure 7.
- Find the total number of circles and squares in Figure 11.
- A figure in the pattern has a total of 421 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 7
= 1 + 7
= 8
(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 11 x 12)
= 1 + 11 x 12
= 1 + 132
= 133
(c)
1 + 2[
12 x Figure Number x (Figure Number + 1)] = 421
2[
12 x Figure Number x (Figure Number + 1)] = 421 - 1
Figure Number x (Figure Number + 1) = 420
20 x 21 = 420
Figure number that has a total number of circles and squares = 20
Answer(s): (a) 8; (b) 133; (c) 20