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

1. The table shows the number of straws used for each figure. What is the number of sticks used for Figure 7?
2. What is the difference in the number of straws Rachel would use for Figure 17 and Figure 19?
3. In which figure number would Rachel use 75 straws?

(a)
Number of straws 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 7 = 20

(b)
Increase in the number of straws that Rachel would use from Figure 5 onwards between two consecutive odd-numbered patterns
= 2 + 3
= 5

Difference in the number of straws that Rachel would use for Figure 17 and Figure 19 = 5

(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.

75 ÷ 2.5 = 30 r 0

Since the remainder is 0, the figure number is an odd number. So, we will apply the formula for the number of sticks for odd-numbered figures.

Figure number x 2.5 + 2.5 = 75
Figure number x 2.5 = 75 - 2.5
Figure number x 2.5 = 72.5
Figure number = 72.5 ÷ 2.5
Figure number = 29

Answer(s): (a) 20; (b) $(n5); (c) 29