Elyse uses rods to form figures that follow a pattern. The first five figures are shown.

1. The table shows the number of rods used for each figure. What is the number of sticks used for Figure 6?
2. What is the difference in the number of rods Elyse would use for Figure 9 and Figure 11?
3. In which figure number would Elyse use 75 rods?

(a)
Number of rods 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 rods that Elyse would use from Figure 5 onwards between two consecutive odd-numbered patterns
= 2 + 3
= 5

Difference in the number of rods that Elyse would use for Figure 9 and Figure 11 = 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) 17; (b) $(n5); (c) 29