1. **State the problem:** We have the arithmetic sequence $-18, -23, -28, -33, \ldots$ and need to find the recursive formula, explicit formula, and the 52nd term. 2. **Identify the common difference:** The difference between consecutive terms is $-23 - (-18) = -5$. 3. **Write the recursive formula:** The first term is $a_1 = -18$ and each term is obtained by subtracting 5 from the previous term. $$a_n = a_{n-1} - 5, \quad a_1 = -18$$ 4. **Write the explicit formula:** The explicit formula for an arithmetic sequence is $$a_n = a_1 + (n-1)d$$ where $d$ is the common difference. Substitute $a_1 = -18$ and $d = -5$: $$a_n = -18 + (n-1)(-5)$$ Simplify: $$a_n = -18 - 5(n-1)$$ $$a_n = -18 - 5n + 5$$ $$a_n = -5n - 13$$ 5. **Find the 52nd term using the explicit formula:** $$a_{52} = -5(52) - 13$$ $$a_{52} = -260 - 13$$ $$a_{52} = -273$$ **Final answers:** - Recursive formula: $a_n = a_{n-1} - 5$, $a_1 = -18$ - Explicit formula: $a_n = -5n - 13$ - 52nd term: $-273$