The diagram shows the first three figures in a sequence of squares with shaded patterns. For each figure, a mathematical pattern is expressed as the following.
Figure 1: 1 + 3 = 4 = 2 x 2 total squares
Figure 2: 1 + 3 + 5 = 9 = 3 x 3 total squares
Figure 3: 1 + 3 + 5 + 7 = 16 = 4 x 4 total squares
- How many white squares are there in Figure 10?
- How many white squares are there in Figure 16?
- Find the sum of 1 + 3 + 5 + 7 + ... + 51.
(a)
Figures |
Number of white squares |
Observe the patterns. |
Create a concise formula |
Figure 1 and 2 |
3 |
4 x 1 - 1 |
4(1) - 1 |
Figure 3 and 4 |
3 + 7 |
4 x 1 - 1 +
4 x 2 - 1
|
4(1 + 2) - 2 |
Figure 5 and 6 |
3 + 7 + 11 |
4 x 1 - 1 +
4 x 2 - 1 +
4 x 3 - 1
|
4(1 + 2 + 3) - 3 |
Formula:
Last number in the bracketed series in even figure number = Figure number ÷ 2
Number of white squares in even figure number
= 4 (1 + 2 + ... + Last number) - Last number
Last number
= Figure Number ÷ 2
= 10 ÷ 2
= 5
Number of white squares in Figure 10
= 4 (1 + 2 + ...+ 5) - 5
= 4 ((5 x 6) ÷ 2) - 5
= 4 x 15 - 5
= 60 - 5
= 55
(b)
Last number
= Figure Number ÷ 2
= 16 ÷ 2
= 8
Number of white squares in Figure 16
= 4 (1 + 2 + ...+ 8) - 8
= 4 [(8 x 9) ÷ 2] - 8
= 4 x 36 - 8
= 144 - 8
= 136
(c)
Sum of consecutive odd numbers = [(1 + Last number) ÷ 2]
2 Sum of consecutive odd from 1 to 51
= [(1 + 51) ÷ 2]
2 = 26
2 = 26 x 26
= 676
Answer(s): (a) 55; (b) 136; (c) 676