Question

which figure is missing in the pattern?

Explanation:

Step1: Analyze the pattern of the figures' dimensions.

Let's assume the first figure has a height of \( h_1 \) and width of \( w_1 \), the second \( h_2, w_2 \), etc.
First figure: Let's count the number of rows (height) and columns (width). First figure: 2 rows, 1 column. Second: 3 rows, 2 columns? Wait, no, looking at the grid:
Wait, first figure: 2 rows (vertical) and 1 column (horizontal). Second: 3 rows, 2 columns? Wait, no, maybe it's the number of squares. Wait, first figure: 2 (2x1), second: 6 (3x2), third: 12 (4x3), fifth: 42 (7x6). Wait, let's check the sequence of the number of squares.
First figure: \( 2 = 2\times1 \)
Second: \( 6 = 3\times2 \)
Third: \( 12 = 4\times3 \)
Fifth: \( 42 = 7\times6 \)
So the pattern for the \( n \)-th figure (where \( n = 1,2,3,4,5 \)): number of squares is \( (n + 1)\times n \)? Wait, no:
For \( n = 1 \): \( (1 + 1)\times1 = 2 \) (matches first figure: 2 squares, 2 rows, 1 column)
\( n = 2 \): \( (2 + 1)\times2 = 6 \) (second figure: 3 rows, 2 columns, 3x2=6, matches)
\( n = 3 \): \( (3 + 1)\times3 = 12 \) (third figure: 4 rows, 3 columns, 4x3=12, matches)
\( n = 5 \): \( (5 + 1)\times5 = 30 \)? Wait, no, the fifth figure looks like 7 rows and 6 columns? Wait, maybe I miscounted. Wait, first figure: 2 rows (vertical) and 1 column (horizontal) → 2 squares. Second: 3 rows, 2 columns → 6 squares. Third: 4 rows, 3 columns → 12 squares. Then the fourth should be 5 rows, 4 columns → 20 squares (5x4=20), and fifth: 6 rows, 5 columns? Wait, no, the fifth figure in the image looks larger. Wait, maybe the pattern is that the number of rows and columns increases by 1 each time.
First figure: rows = 2, columns = 1
Second: rows = 3, columns = 2
Third: rows = 4, columns = 3
So the fourth figure should have rows = 5, columns = 4 (since each time rows = previous rows +1, columns = previous columns +1). Let's check:
First: 2 rows, 1 column (2x1)
Second: 3 rows, 2 columns (3x2)
Third: 4 rows, 3 columns (4x3)
Fourth: 5 rows, 4 columns (5x4)
Fifth: 6 rows, 5 columns? Wait, but the fifth figure in the image looks like 7 rows and 6 columns. Wait, maybe my initial count is wrong. Let's look at the figures again:

First figure: small, 2 rows (vertical) and 1 column (horizontal) → 2 squares.

Second: 3 rows, 2 columns → 6 squares (3x2).

Third: 4 rows, 3 columns → 12 squares (4x3).

Fifth: 7 rows, 6 columns → 42 squares (7x6). Wait, so the sequence of rows: 2, 3, 4,?, 7. So the rows are increasing by 1 each time: 2,3,4,5,6,7? Wait, no, 2 to 3 ( +1), 3 to 4 (+1), 4 to 5 (+1), 5 to 6 (+1), 6 to 7 (+1). So the fourth figure should have 5 rows and 4 columns (since columns: 1,2,3,?,6. Wait, columns: 1,2,3,4,5,6? Wait, first column 1, second 2, third 3, fourth 4, fifth 5, sixth 6. So rows: 2,3,4,5,6,7. So the fourth figure (position 4) should have rows = 5, columns = 4. So the figure with 5 rows (vertical) and 4 columns (horizontal), i.e., a grid of 5x4 squares.

Step2: Confirm the pattern.

The pattern of the number of rows (r) and columns (c) for each figure (index i, starting at 1):
- For i=1: r=2, c=1 (r = i +1, c = i)
- i=2: r=3, c=2 (r = i +1, c = i)
- i=3: r=4, c=3 (r = i +1, c = i)
- i=4: r=5, c=4 (r = i +1, c = i)
- i=5: r=6, c=5? Wait, but the fifth figure in the image looks like 7 rows and 6 columns. Wait, maybe I made a mistake. Wait, maybe the first figure is 1x2 (rows x columns)? No, the first figure is taller than wide. Wait, vertical rows: first figure has 2 vertical cells (rows) and 1 horizontal cell (column). So rows (vertical) = 2, columns (horizontal) = 1. Second: rows=3, columns=2. Third: rows=4, columns=3. So the fourth should be rows=5, columns=4. So the missing figure is a grid with 5 rows (vertical) and 4 columns (horizontal), i.e., 5x4 grid of squares.

Answer:

The missing figure is a grid with 5 rows (vertical) and 4 columns (horizontal) (a 5×4 grid of blue squares, following the pattern where each subsequent figure increases the number of rows and columns by 1 from the previous one).