1. **State the problem:** We are given the sequence 3, 7, 11, 15, 19, … and need to determine which given explicit or recursive definitions correctly describe the n-th term of this sequence. 2. **Identify the pattern:** The sequence increases by 4 each time: 7 - 3 = 4, 11 - 7 = 4, etc. This is an arithmetic sequence with first term $a_1 = 3$ and common difference $d = 4$. 3. **General formula for arithmetic sequences:** The explicit formula is $$f(n) = a_1 + (n - 1)d$$ where $a_1$ is the first term and $d$ is the common difference. 4. **Apply the formula:** $$f(n) = 3 + (n - 1) \times 4 = 3 + 4n - 4 = 4n - 1$$ 5. **Check each option:** - $f(n) = 3 + 4(n - 1)$ for $n \geq 1$ simplifies to $4n - 1$, which matches the sequence. **Correct**. - Recursive: $f(n) = 3$, $f(n+1) = f(n) + 4$ for $n \geq 1$ means start at 3 and add 4 each time, matching the sequence. **Correct**. - Recursive: $f(n) = 1$, $f(n+1) = f(n) + 3$ for $n > 1$ starts at 1 and adds 3, which does not match the sequence. **Incorrect**. - $f(n) = 3 + 4n$ for $n \geq 1$ equals $4n + 3$, which does not match the sequence values. **Incorrect**. - Recursive: $f(1) = 3$, $f(n+1) = 4(f(n))$ multiplies by 4 each time, which grows too fast and does not match. **Incorrect**. - $f(n) = 4 + 3n$ for $n \geq 1$ equals $3n + 4$, which does not match the sequence. **Incorrect**. **Final answer:** The correct definitions are: - $f(n) = 3 + 4(n - 1)$ for all $n \geq 1$ - $f(n) = 3$, $f(n + 1) = f(n) + 4$ for all $n \geq 1$