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

1. How many white squares are there in Figure 8?
2. How many white squares are there in Figure 14?
3. Find the sum of 1 + 3 + 5 + 7 + ... + 33.

(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
= 8 ÷ 2
= 4

Number of white squares in Figure 8
= 4 (1 + 2 + ...+ 4) - 4
= 4 ((4 x 5) ÷ 2) - 4
= 4 x 10 - 4
= 40 - 4
= 36

(b)
Last number
= Figure Number ÷ 2
= 14 ÷ 2
= 7

Number of white squares in Figure 14
= 4 (1 + 2 + ...+ 7) - 7
= 4 [(7 x 8) ÷ 2] - 7
= 4 x 28 - 7
= 112 - 7
= 105

(c)
Sum of consecutive odd numbers = [(1 + Last number) ÷ 2]²

Sum of consecutive odd from 1 to 33
= [(1 + 33) ÷ 2]²
= 17²
= 17 x 17
= 289

Answer(s): (a) 36; (b) 105; (c) 289