1. **Problem Statement:** Given the arithmetic sequence $-15, -7, 1, 9, \ldots$, write the recursive formula, explicit formula, and find the 52nd term using the explicit formula. 2. **Identify the first term and common difference:** - The first term $a_1 = -15$. - The common difference $d$ is found by subtracting the first term from the second term: $$d = -7 - (-15) = -7 + 15 = 8$$ 3. **Write the recursive formula:** The recursive formula for an arithmetic sequence is: $$a_n = a_{n-1} + d$$ with the initial term: $$a_1 = -15$$ So, $$a_n = a_{n-1} + 8, \quad a_1 = -15$$ 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 $a_1 = -15$ and $d = 8$: $$a_n = -15 + (n-1) \times 8$$ Simplify: $$a_n = -15 + 8n - 8 = 8n - 23$$ 5. **Find the 52nd term using the explicit formula:** $$a_{52} = 8 \times 52 - 23$$ Calculate: $$a_{52} = 416 - 23 = 393$$ **Final answers:** - Recursive formula: $a_n = a_{n-1} + 8$, $a_1 = -15$ - Explicit formula: $a_n = 8n - 23$ - 52nd term: $393$