The first four figures of a pattern as shown.

The table shows the number of dots and the number of non-overlapping triangles for each figure.

1. Find the values of (i) and (ii). Give your answers in the following format. (Eg 1, 2)
2. What is the number of dots in Figure 13?
3. In which Figure number will there be 196 dots?

(a)
(i)
Number of dots:
Figure 1: 6 = 1 + 1 x 5
Figure 2: 11 = 1 + 2 x 5
Figure 3: 16 = 1 + 3 x 5
Figure 4: 21 = 1 + 4 x 5

Formula:
Number of dots = 1 + Figure Number x 5

Number of dots in Figure 5
= 1 + 5 x 5
= 1 + 25
= 26

(ii)
Number of non-overlapping triangles:
Figure 1:   5 = 5 x 1
Figure 2: 10 = 5 x 2
Figure 3: 20 = 5 x 4 (Notice that '3' is skipped)
Figure 4: 25 = 5 x 5
Figure 5: 35 = 5 x 7 (Notice that 6 is skipped)
Figure 6: 40 = 5 x 8

Number of non-overlapping triangles for each figure is a multiple of 5.
So, the number '5' is one of the factors.

The pattern of the other factor in each equation is as follows:
1, 2, (skip 3) 4, 5, (skip 6) 7, 8, (skip 9) 10, 11, (skip 12) 13, 14, (skip 15) 16, 17

Number of overlapping triangles in Figure 5
= 7 x 5
= 35

(b)
Number of dots in Figure 13
= 1 + 13 x 5
= 1 + 65
= 66

(c)
1 + Figure Number x 5 = 196
Figure Number x 5 = 196 - 1
Figure Number x 5 = 195
Figure Number = 195 ÷ 5
Figure Number = 39

Answer(s): (a) 26, 35; (b) 66; (c) 39