1. **State the problem:** We have an arithmetic sequence starting with 4, -1, -6, -11, ... 2. **Identify the common difference:** The difference between consecutive terms is $-1 - 4 = -5$. 3. **Write the recursive formula:** The recursive formula for an arithmetic sequence is $a_n = a_{n-1} + d$ with $a_1$ as the first term. Here, $a_1 = 4$ and $d = -5$, so: $$a_n = a_{n-1} - 5$$ 4. **Write the explicit formula:** The explicit formula for the $n$th term of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ Substitute $a_1 = 4$ and $d = -5$: $$a_n = 4 + (n-1)(-5) = 4 - 5(n-1)$$ 5. **Find the 52nd term using the explicit formula:** $$a_{52} = 4 - 5(52-1) = 4 - 5 \times 51$$ $$a_{52} = 4 - 255 = -251$$ **Final answers:** - Recursive formula: $a_n = a_{n-1} - 5$ with $a_1 = 4$ - Explicit formula: $a_n = 4 - 5(n-1)$ - 52nd term: $a_{52} = -251$