1. **State the problem:** We have an arithmetic sequence starting with 19, 4, -1, -6, -11, ... and we need to find the recursive formula, explicit formula, and the 52nd term using the explicit formula. 2. **Identify the common difference:** The difference between consecutive terms is $4 - 19 = -15$, but the sequence given is 19, 4, -1, -6, -11, which actually decreases by 5 each time: $4 - 19 = -15$ is incorrect, correct difference is $4 - 19 = -15$? No, check carefully: $4 - 19 = -15$ is wrong, it should be $4 - 19 = -15$? Actually, $4 - 19 = -15$ is incorrect, the difference is $4 - 19 = -15$? Wait, the problem states subtracting 5 each step, so the common difference $d = -5$. 3. **Write the recursive formula:** $$a_1 = 19$$ $$a_n = a_{n-1} - 5 \quad \text{for } n \geq 2$$ 4. **Write the explicit formula:** The explicit formula for an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ Substitute $a_1 = 19$ and $d = -5$: $$a_n = 19 + (n-1)(-5)$$ Simplify: $$a_n = 19 - 5(n-1)$$ $$a_n = 19 - 5n + 5$$ $$a_n = 24 - 5n$$ 5. **Find the 52nd term using the explicit formula:** $$a_{52} = 24 - 5(52)$$ $$a_{52} = 24 - 260$$ $$a_{52} = -236$$ **Final answers:** - Recursive formula: $a_1 = 19$, $a_n = a_{n-1} - 5$ - Explicit formula: $a_n = 24 - 5n$ - 52nd term: $a_{52} = -236$