Question 20

We will identify the growth pattern for each figure by counting the basic units in each step and then use these patterns to find the next figure, the number of units in Step 10, and a general expression.

Step 1 — Analyze the yellow pattern

Let us count the basic units in each step for the yellow pattern. We can see the figure is made of three parts: a bottom vertical column, a middle horizontal row, and a top vertical column.

For the bottom vertical column: Step 1 has 3 units. Step 2 has 5 units. Step 3 has 7 units. This is an arithmetic progression where each step adds 2 units. The number of units in Step $n$ is $2n+1$.

For the middle horizontal row: Step 1 has 2 units. Step 2 has 4 units. Step 3 has 6 units. This is an arithmetic progression where each step adds 2 units. The number of units in Step $n$ is $2n$.

For the top vertical column: Step 1 has 2 units. Step 2 has 6 units. Step 3 has 14 units. Let us find the differences between consecutive terms: $6-2=4$, $14-6=8$. The second differences are $8-4=4$. This indicates a quadratic pattern of the form $An^2 + Bn + C$. We find $A=2$, $B=-2$, $C=2$. So, the number of units in Step $n$ is $2n^2 - 2n + 2$.

Now, let us find the total number of units for Step $n$ by adding the units from all three parts: $S_n = (2n+1) + (2n) + (2n^2 - 2n + 2)$ $S_n = 2n^2 + (2n + 2n - 2n) + (1 + 2)$ $S_n = 2n^2 + 2n + 3$ Let us verify this formula: For Step 1: $S_1 = 2(1)^2 + 2(1) + 3 = 2 + 2 + 3 = \mathbf{7}$. For Step 2: $S_2 = 2(2)^2 + 2(2) + 3 = 8 + 4 + 3 = \mathbf{15}$. For Step 3: $S_3 = 2(3)^2 + 2(3) + 3 = 18 + 6 + 3 = \mathbf{27}$. The formula is correct.

Step 2 — Yellow pattern: Draw Step 4

To draw Step 4, we use the formulas for each part: The bottom vertical column will have $2(4)+1 = \mathbf{9}$ units. The middle horizontal row will have $2(4) = \mathbf{8}$ units. The top vertical column will have $2(4)^2 - 2(4) + 2 = 32 - 8 + 2 = \mathbf{26}$ units. The total units in Step 4 will be $9 + 8 + 26 = \mathbf{43}$ units.

Step 3 — Yellow pattern: Units in Step 10

We use the formula $S_n = 2n^2 + 2n + 3$ for $n=10$: $S_{10} = 2(10)^2 + 2(10) + 3$ $= 2(100) + 20 + 3$ $= 200 + 20 + 3$

$\boxed{223 \text{ units}}$

Step 4 — Yellow pattern: Expression for Step y

We replace $n$ with $y$ in our formula for the total number of units:

$\boxed{S_y = 2y^2 + 2y + 3}$

Step 5 — Analyze the blue pattern

Let us count the basic units in each step for the blue pattern. We can see the figure is a rectangle of squares with one square missing from the bottom-right corner.

For Step 1: The figure is a $2 \times 3$ rectangle with 1 square missing. Number of units = $(2 \times 3) - 1 = 6 - 1 = \mathbf{5}$.

For Step 2: The figure is a $3 \times 4$ rectangle with 1 square missing. Number of units = $(3 \times 4) - 1 = 12 - 1 = \mathbf{11}$.

For Step 3: The figure is a $4 \times 5$ rectangle with 1 square missing. Number of units = $(4 \times 5) - 1 = 20 - 1 = \mathbf{19}$.

We observe that for Step $n$, the rectangle has $(n+1)$ rows and $(n+2)$ columns. The number of units in Step $n$ is $(n+1)(n+2) - 1$. Let us expand this expression: $S_n = (n+1)(n+2) - 1$ $= (n^2 + 2n + n + 2) - 1$ $= n^2 + 3n + 2 - 1$ $= n^2 + 3n + 1$ Let us verify this formula: For Step 1: $S_1 = 1^2 + 3(1) + 1 = 1 + 3 + 1 = \mathbf{5}$. For Step 2: $S_2 = 2^2 + 3(2) + 1 = 4 + 6 + 1 = \mathbf{11}$. For Step 3: $S_3 = 3^2 + 3(3) + 1 = 9 + 9 + 1 = \mathbf{19}$. The formula is correct.

Step 6 — Blue pattern: Draw Step 4

To draw Step 4, we use the pattern for the rectangle dimensions: The rectangle will have $(4+1) = \mathbf{5}$ rows and $(4+2) = \mathbf{6}$ columns. One square will be missing from the bottom-right corner. The total units in Step 4 will be $(5 \times 6) - 1 = 30 - 1 = \mathbf{29}$ units.

Step 7 — Blue pattern: Units in Step 10

We use the formula $S_n = n^2 + 3n + 1$ for $n=10$: $S_{10} = 10^2 + 3(10) + 1$ $= 100 + 30 + 1$

$\boxed{131 \text{ units}}$

Step 8 — Blue pattern: Expression for Step y

We replace $n$ with $y$ in our formula for the total number of units:

$\boxed{S_y = y^2 + 3y + 1}$

Answer

(i) For the yellow pattern, Step 4 consists of a bottom vertical column of 9 squares, a middle horizontal row of 8 squares extending from the top of the bottom column, and a top vertical column of 26 squares extending upwards from the end of the horizontal row. For the blue pattern, Step 4 is a $5 \times 6$ grid of squares with the square at the bottom-right corner removed. (ii) For the yellow pattern, there are 223 basic units in Step 10. For the blue pattern, there are 131 basic units in Step 10. (iii) For the yellow pattern, the expression is $\mathbf{2y^2 + 2y + 3}$. For the blue pattern, the expression is $\mathbf{y^2 + 3y + 1}$.