The graph shows 3 figures formed by shaded and white triangles.
Pattern number |
1 |
2 |
3 |
... |
8 |
Number of shaded triangles |
0 |
2 |
5 |
... |
(i) |
Total number of triangles |
1 |
3 |
8 |
... |
(ii) |
- Complete the column for Figure 8. Give your answers for (i) and (ii) in numbers. (Eg 1, 2)
- What is the total number of triangles in Figure 18?
- How many shaded triangles are there in Figure 21?
(a)
Pattern for the number of shaded triangles
Figure 1: 0 =
0 =
1 - 1Figure 2: 0 + 2 =
2 =
1 + 2 - 1Figure 3: 0 + 2 + 3 =
5 =
1 + 2 + 3 - 1Formula:
Number of shaded triangles =
Sum of all numbers to Figure number - 1Number of shaded triangles in Figure 8
= (1 + 2 + 3 + ... + 8) - 1
= (8 x 9 ÷ 2) - 1
= 36 - 1
= 35 (i)
Pattern for the number of triangles from Figure 2 onwards.
Figure 2:
22 - 1 = 2 x 2 - 1 =
3 Figure 3:
32 - 1 = 3 x 3 - 1 =
8 Formula:
Number of triangles =
Figure number2 - 1 Total number of triangles in Figure 8
= 8
2 - 1
= 8 x 8 - 1
= 64 - 1
= 63 (ii)
(b)
Total number of triangles in Figure 18
= 18
2 - 1
= 18 x 18 - 1
= 324 - 1
= 323
(c)
Number of shaded triangles in Figure 21
= (1 + 2 + 3 + ... + 21) - 1
= (21 x 22 ÷ 2) - 1
= 231 - 1
= 230
Answer(s): (a) 35, 63; (b) 323; (c) 230