1. **Problem statement:** Given the arithmetic sequence $6, 206, 406, 606, \ldots$, write the recursive formula, explicit formula, and find the 52nd term using the explicit formula. 2. **Identify the common difference:** The difference between consecutive terms is $206 - 6 = 200$. 3. **Recursive formula:** The recursive formula for an arithmetic sequence is $a_n = a_{n-1} + d$ with the first term $a_1$ given. Here, $a_1 = 6$ and $d = 200$, so: $$a_n = a_{n-1} + 200, \quad a_1 = 6$$ 4. **Explicit formula:** The explicit formula for the $n$th term of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ Substitute $a_1 = 6$ and $d = 200$: $$a_n = 6 + (n-1) \times 200$$ 5. **Find the 52nd term:** Substitute $n=52$ into the explicit formula: $$a_{52} = 6 + (52-1) \times 200$$ $$a_{52} = 6 + 51 \times 200$$ $$a_{52} = 6 + 10200$$ $$a_{52} = 10206$$ **Final answers:** - Recursive formula: $a_n = a_{n-1} + 200$, $a_1 = 6$ - Explicit formula: $a_n = 6 + (n-1) \times 200$ - 52nd term: $10206$