PART A: Use the given arithmetic sequence to write an equation. \[ 8,2,-4,-10, \ldots \] PART B: Find the 35 th term.

Sure, let's break down the problem into two parts: ### **PART A: Writing the Equation of the Arithmetic Sequence** An arithmetic sequence is defined by its first term and the common difference between consecutive terms. The general formula for the \( n \)-th term (\( a_n \)) of an arithmetic sequence is: \[ a_n = a_1 + (n - 1)d \] Where: - \( a_1 \) is the first term. - \( d \) is the common difference. - \( n \) is the term number. **Given Sequence:** \( 8, 2, -4, -10, \ldots \) 1. **Identify the First Term (\( a_1 \))**: \[ a_1 = 8 \] 2. **Determine the Common Difference (\( d \))**: \[ d = a_2 - a_1 = 2 - 8 = -6 \] 3. **Write the Equation:** Plugging the values into the general formula: \[ a_n = 8 + (n - 1)(-6) \] Simplifying: \[ a_n = 8 - 6(n - 1) = 8 - 6n + 6 = 14 - 6n \] **Final Equation:** \[ a_n = 14 - 6n \] Alternatively, you can leave it in the first form: \[ a_n = 8 + (n - 1)(-6) \] ### **PART B: Finding the 35th Term** Using the equation derived in Part A: \[ a_{35} = 8 + (35 - 1)(-6) = 8 + 34 \times (-6) \] Calculate the multiplication: \[ 34 \times (-6) = -204 \] Add this to the first term: \[ a_{35} = 8 - 204 = -196 \] **The 35th term of the sequence is \(-196\).** --- **Summary:** - **Part A:** The equation of the sequence is \( a_n = 8 + (n - 1)(-6) \) or \( a_n = 14 - 6n \). - **Part B:** The 35th term is \(-196\).