The pattern shows a sequence of triangles constructed using rods. The total number of small triangles formed and the total number of rods used for the first three patterns are shown on the table.
- Find the number of small triangles in Figure 9.
- Find the number of small triangles in Figure 20.
- Find the number of rods in Figure 41.
Figure |
1 |
2 |
3 |
9 |
20 |
41 |
Number of small triangles |
1 |
4 |
9 |
(a) ? |
(b) ? |
|
Number of rods |
3 |
9 |
18 |
... |
|
(c) ? |
(a)
Pattern for the number of small triangles
Figure 1: 1
2 = 1
Figure 2: 2
2 = 4
Figure 3: 3
2 = 9
Formula
Number of small triangles = Figure number
2 Number of small triangles for Figure 9
= 9
2 = 81
(b)
Number of small triangles for Figure 20
= 20
2 = 20 x 20
= 400
(c)
Patterns for the number of rods
Figure 1: 3 x (1)
Figure 2: 3 x (1 + 2)
Figure 3: 3 x (1 + 2 + 3)
Formula
Number of rods = 3 x [Figure Number x (Figure number + 1) ÷ 2]
Number of rods in Figure 41
= 3 x (41 x (41+1) ÷ 2)
= 3 x (41 x 42 ÷ 2)
= 3 x (1722 ÷ 2)
= 3 x 861
= 2583
Answer(s): (a) 81; (b) 400; (c) 2583