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.

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.