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kindly answer this math question correctly

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### Part A: Identify the next figure in the following groups of shapes.

**1.**
The sequence consists of circles inside squares, with the black half of the circle rotating by 90° in each figure. The rotation occurs in the clockwise direction.

- **Answer:** The next figure will be a square containing a circle where the black half of the circle is at the bottom-left.

**2.**
In this sequence, a dot alternates between different positions on a grid. The dot moves along a pattern.

- **Answer:** The dot moves to the bottom-left in the next figure, following the logical progression.

**3.**
Here, the sequence involves arrows and bars. It appears that the number of arrows or bars may be changing systematically.

- **Answer:** The next figure will involve more arrows or changes based on the rule governing the sequence.

### Part B: Solve the following sequence and write the arithmetic formula to find the correct answer.

1. **What is the 10th term of the sequence 1, 11, 21, 31, 41, 51, 61, 71, 81, ...?**
   - **Answer:** This is an arithmetic sequence where each term increases by 10.
     - Formula: \( a_n = 1 + (n-1) \times 10 \)
     - For \( n = 10 \): \( a_{10} = 1 + (10 - 1) \times 10 = 1 + 90 = 91 \)
     - **The 10th term is 91.**

2. **The sum of three consecutive even numbers is 48. What are the three numbers?**
   - Let the three numbers be \( x, x+2, x+4 \).
   - The equation becomes \( x + (x+2) + (x+4) = 48 \).
   - Simplifying: \( 3x + 6 = 48 \)
   - Subtract 6 from both sides: \( 3x = 42 \)
   - Divide by 3: \( x = 14 \)
   - Therefore, the numbers are 14, 16, and 18.
   - **The three numbers are 14, 16, and 18.**

3. **The product of two numbers is 56 and their sum is 15. What are the two numbers?**
   - Let the numbers be \( x \) and \( y \).
   - The system of equations is:
     1. \( x + y = 15 \)
     2. \( x \times y = 56 \)
   - Solve the quadratic equation: \( x^2 - 15x + 56 = 0 \)
   - Using the quadratic formula:
     - \( x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(1)(56)}}{2(1)} = \frac{15 \pm \sqrt{225 - 224}}{2} = \frac{15 \pm 1}{2} \)
     - Solutions: \( x = 8 \) or \( x = 7 \)
   - The numbers are 7 and 8.
   - **The two numbers are 7 and 8.**

4. **In the figure below, how many small squares are there?**
   - It seems to be asking for the number of smaller units in a larger grid. Let's assume it's a 3x3 or 4x4 grid.
   - **Answer:** Assuming it's a 3x3 grid, there would be 9 squares. (Please confirm based on the figure.)

5. **What is the simplified value of the expression \( (2x + 7)(2x - 3) \)?**
   - Use the distributive property (FOIL):
     - \( (2x + 7)(2x - 3) = 4x^2 - 6x + 14x - 21 \)
     - Combine like terms: \( 4x^2 + 8x - 21 \)
   - **The simplified expression is \( 4x^2 + 8x - 21 \).**

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Would you like more details or have any questions on these solutions? 

Here are some relative questions for further practice:
1. How would the sequence change if the increment were by 5 instead of 10 in question B.1?
2. What if the sum of four consecutive even numbers were 60? What would the numbers be?
3. How would the quadratic formula change if the product of two numbers were 72 and their sum 18?
4. How many smaller squares would be present in a 4x4 grid instead of a 3x3 grid?
5. How do you expand \( (3x + 2)(2x - 5) \)?

**Tip:** For arithmetic sequences, always check the pattern of differences between terms.

Explore detailed solutions and step-by-step explanations for math problems, including arithmetic sequences, quadratic equations, and the distributive property (FOIL). Perfect for students in Grades 7-9.

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