Penelope 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 Penelope would use for Figure 13 and Figure 15?
3. In which figure number would Penelope use 50 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 Penelope would use from Figure 5 onwards between two consecutive odd-numbered patterns
= 2 + 3
= 5

Difference in the number of sticks that Penelope would use for Figure 13 and Figure 15 = 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.

50 ÷ 2.5 = 20 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 = 50
Figure number x 2.5 = 50 - 2.5
Figure number x 2.5 = 47.5
Figure number = 47.5 ÷ 2.5
Figure number = 19

Answer(s): (a) 17; (b) $(n5); (c) 19