Find a formula for the \( n \) nh term in this arithmetic sequence: \( a_{1}=28, a_{2}=21, a_{3}=14, a_{4}=7, \ldots \) \[ a_{n}=[?] n+[] \] Hint: The coefficient of \( n \) is the common difference.

To find the formula for the \( n \)-th term of the given arithmetic sequence: \[ a_1 = 28, \ a_2 = 21, \ a_3 = 14, \ a_4 = 7, \ldots \] **Step 1: Determine the Common Difference (\( d \))** The common difference \( d \) in an arithmetic sequence is the difference between any two consecutive terms. \[ d = a_2 - a_1 = 21 - 28 = -7 \] **Step 2: Use the General Formula for an Arithmetic Sequence** The general formula for the \( n \)-th term of an arithmetic sequence is: \[ a_n = a_1 + (n - 1)d \] Substituting the known values: \[ a_n = 28 + (n - 1)(-7) \] \[ a_n = 28 - 7n + 7 \] \[ a_n = -7n + 35 \] **Final Formula:** \[ a_n = -7n + 35 \] **Verification:** Let's verify the formula with the given terms: - For \( n = 1 \): \[ a_1 = -7(1) + 35 = 28 \] - For \( n = 2 \): \[ a_2 = -7(2) + 35 = 21 \] - For \( n = 3 \): \[ a_3 = -7(3) + 35 = 14 \] - For \( n = 4 \): \[ a_4 = -7(4) + 35 = 7 \] All terms match the given sequence. **Answer:** After simplifying, the nth term is –7 · n plus 35. Thus, aₙ = −7 n + 35