Rachel uses sticks to form figures that follow a pattern. The first five figures are shown.

1. The table shows the number of sticks used for each figure. What is the number of sticks used for Figure 6?
2. What is the difference in the number of sticks Rachel would use for Figure 11 and Figure 12?
3. In which figure number would Rachel use 57 sticks?

(a)
Number of sticks used:
Figure 1: 1 + 3 = 4
Figure 2: 1 + 3 + 3 = 7
Figure 3: 1 + 3 + 3 + 3 = 10
Figure 4: 1 + 3 + 3 + 3 + 2 = 12
Figure 5: 1 + 3 + 3 + 3 + 2 + 3 = 15
Figure 6: 1 + 3 + 3 + 3 + 2 + 3 + 2 = 17
Figure 7: 1 + 3 + 3 + 3 + 2 + 3 + 2 + 3 = 20

Number of sticks used for Figure 6 = 17

(b)
Increase in the number of sticks that Rachel would use from Figure 5 onwards between one odd-numbered pattern to the next consecutive even-numbered pattern = 2

Difference in the number of sticks that Rachel would use between Figure 11 and Figure 12 = 2

(c)
Number of sticks used for even-numbered figures from Figure 4 onwards:
Figure 4: 12 = 4 x 2.5 + 2
Figure 6: 17 = 6 x 2.5 + 2

Formula:
Number of sticks used for even-numbered figures from Figure 4 onwards = Figure number x 2.5 + 2

Number of sticks used for odd-numbered figures from Figure 5 onwards:
Figure 5: 15 = 5 x 2.5 + 2.5
Figure 7: 20 = 7 x 2.5 + 2.5

Formula:
Number of sticks used for odd-numbered figures from Figure 5 onwards = Figure number x 2.5 + 2.5

Number of sticks used for even-numbered figures from Figure 4 onwards has a remainder of 2 when divided by 2.5.

Number of sticks used for odd-numbered figures from Figure 5 onwards has no remainder when divided by 2.5.

57 ÷ 2.5 = 22 r 2

Since the remainder is 2, the figure number is an even number. So, we will apply the formula for the number of sticks for even-numbered figures.

Figure number x 2.5 + 2 = 57
Figure number x 2.5 = 57 - 2
Figure number x 2.5 = 55
Figure number = 55 ÷ 2.5
Figure number = 22

Answer(s): (a) 17; (b) 2; (c) 22