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I see the image contains multiple math problems, specifically focused on recursive sequences and their related terms. Let me help you with each one.

### Problem 6: 
"Find the first four terms of the recursive sequence defined below."
- \( a_1 = -3 \)
- \( a_n = a_{n-1} - n \)

**Solution:**
- \( a_1 = -3 \)
- \( a_2 = a_1 - 2 = -3 - 2 = -5 \)
- \( a_3 = a_2 - 3 = -5 - 3 = -8 \)
- \( a_4 = a_3 - 4 = -8 - 4 = -12 \)

**Answer:**
The first four terms are \( a_1 = -3 \), \( a_2 = -5 \), \( a_3 = -8 \), and \( a_4 = -12 \).

### Problem 7:
"A sequence is defined recursively by \( f(1) = 16 \) and \( f(n) = f(n-1) + 2n \). Find \( f(4) \)."

**Solution:**
- \( f(1) = 16 \)
- \( f(2) = f(1) + 2(2) = 16 + 4 = 20 \)
- \( f(3) = f(2) + 2(3) = 20 + 6 = 26 \)
- \( f(4) = f(3) + 2(4) = 26 + 8 = 34 \)

**Answer:**
\( f(4) = 34 \).

### Problem 8:
"Find the third term in the recursive sequence \( a_{n+1} = 2a_n - 1 \), where \( a_1 = 3 \)."

**Solution:**
- \( a_1 = 3 \)
- \( a_2 = 2a_1 - 1 = 2(3) - 1 = 6 - 1 = 5 \)
- \( a_3 = 2a_2 - 1 = 2(5) - 1 = 10 - 1 = 9 \)

**Answer:**
\( a_3 = 9 \).

### Problem 9:
"Which recursively defined function represents the sequence 3, 7, 15, 31...?"
Looking at the recursive pattern in the sequence:
- \( 7 = 2(3) + 1 \)
- \( 15 = 2(7) + 1 \)
- \( 31 = 2(15) + 1 \)

**Answer:**
Option 3: \( f(1) = 3, \, f(n+1) = 2f(n) + 1 \).

### Problem 10:
"What is the fourth term of the sequence defined by \( a_1 = 3xy^5 \) and \( a_n = \left( \frac{2x}{y} \right) a_{n-1} \)?"

**Solution:**
Start with \( a_1 = 3xy^5 \).
- \( a_2 = \left( \frac{2x}{y} \right) a_1 = \frac{2x}{y} \times 3xy^5 = 6x^2y^4 \)
- \( a_3 = \left( \frac{2x}{y} \right) a_2 = \frac{2x}{y} \times 6x^2y^4 = 12x^3y^3 \)
- \( a_4 = \left( \frac{2x}{y} \right) a_3 = \frac{2x}{y} \times 12x^3y^3 = 24x^4y^2 \)

**Answer:**
The fourth term is \( 24x^4y^2 \), which corresponds to Option 3.

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Would you like further details on any of these solutions, or do you have additional questions? Here are 5 related questions:

1. How would the recursive formula change if the sequence in problem 6 was defined as \( a_n = a_{n-1} + n \)?
2. What is the general formula for the sequence defined in problem 7?
3. How would the solution to problem 9 change if the sequence started with 5 instead of 3?
4. Could the formula for problem 10 work for negative exponents in \( a_1 \)?
5. How would the solution to problem 8 change if \( a_1 \) was 4 instead of 3?

**Tip:** When working with recursive sequences, calculating a few initial terms helps to identify patterns and verify formulas.

Find the first four terms of the recursive sequence defined below. a1 = -3 and an = an-1 - n. Also, several other recursive sequence questions, including a defined sequence with f(1) = 16, determining a function for a given sequence, and finding terms of specific recursive sequences.

Learn how to solve recursive sequences including arithmetic and geometric progressions. Problems include defining recursive terms, applying formulas, and identifying patterns. Suitable for students in Grades 9-12.

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