Question

consider the pattern.
represent the number of shaded squares in each of the three figures as a numeric sequence.
then, complete the table of values.

enter the answer in each space provided. use numbers instead of words.

□, □, □

Explanation:

Step1: Analyze first figure

Count shaded squares: 8 (outer 8, inner 0? Wait, first figure: 3x3 square with center 1 unshaded. So total squares \(3\times3 = 9\), unshaded 1, so shaded \(9 - 1 = 8\)? Wait no, first figure: let's see, first square: 3x3, center 1 white, so shaded: 8? Wait second figure: 4x4? No, second figure: outer layer, maybe 4x4 with inner 2x2 unshaded? Wait no, first figure: 3x3 (side length 3), shaded: \(3^2 - 1^2 = 9 - 1 = 8\). Second figure: side length 4? Wait no, first figure: side 3, second side 4? Wait no, first figure: 3x3, shaded: 8 (3² -1²). Second figure: 4x4? No, second figure: 4x4 with inner 2x2 unshaded? Wait 4² - 2² = 16 - 4 = 12? Wait third figure: 5x5 - 3² = 25 - 9 = 16? Wait let's count:

First figure: 3x3 grid, center 1 square unshaded. So shaded: 9 - 1 = 8.

Second figure: 4x4 grid? No, wait the second figure: outer is 4x4? Wait no, first figure: 3x3 (rows and columns: 3), second: 4x4? Wait no, looking at the figures:

First figure: 3x3 (3 rows, 3 columns), center 1 unshaded. Shaded: 8.

Second figure: 4x4? No, wait the second figure has a larger square, inner square is 2x2 unshaded. So 4x4 - 2x2 = 16 - 4 = 12.

Third figure: 5x5 - 3x3 = 25 - 9 = 16. Wait let's check:

First figure: 3x3, shaded: 8 (3² -1² = 8).

Second figure: 4x4? No, wait the side length increases by 1 each time? Wait first side: 3, second: 4, third: 5? No, wait the inner square: first inner 1x1, second inner 2x2, third inner 3x3. So outer square: first 3x3 (inner 1x1), second 4x4 (inner 2x2), third 5x5 (inner 3x3). So shaded:

First: \(3^2 - 1^2 = 9 - 1 = 8\)

Second: \(4^2 - 2^2 = 16 - 4 = 12\)

Third: \(5^2 - 3^2 = 25 - 9 = 16\)

Wait but let's count manually. First figure: 3x3, center 1 white, so 8 shaded. Second figure: 4x4? Wait no, the second figure: let's count the shaded squares. The outer layer: top row 4, bottom row 4, left column (excluding top and bottom) 2, right column (excluding top and bottom) 2. Wait 4 + 4 + 2 + 2 = 12. Yes, that's 12. Third figure: top row 5, bottom row 5, left column (excluding top and bottom) 3, right column (excluding top and bottom) 3. 5 + 5 + 3 + 3 = 16. Yes. So the sequence is 8, 12, 16.

Step2: Confirm each term

First term: 8 (from 3x3 - 1x1)

Second term: 12 (from 4x4 - 2x2)

Third term: 16 (from 5x5 - 3x3)

Answer:

8, 12, 16