Which of the following is not an accurate formula to describe the following arithmetic sequence? \[ \begin{array}{l}78,72,66,60,54 \ldots \\ a_{n}=-6 n+90 \\ a_{n}=78-6(n-1) \\ a_{n}=84-6 n \\ a_{n}=-6 n+84\end{array} \]

Let's analyze each formula to determine which one does **not** accurately describe the arithmetic sequence \(78, 72, 66, 60, 54, \ldots\). ### Given: - **First Term (\(a_1\))**: 78 - **Common Difference (\(d\))**: \(-6\) The general formula for the \(n\)-th term of an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \] Substituting the known values: \[ a_n = 78 + (n-1)(-6) = 78 - 6(n-1) \] ### Evaluating Each Option: 1. **\(a_n = -6n + 90\)** - **For \(n = 1\):** \[ a_1 = -6(1) + 90 = 84 \] - **Expected:** 78 - **Result:** **Incorrect** 2. **\(a_n = 78 - 6(n-1)\)** - **For \(n = 1\):** \[ a_1 = 78 - 6(0) = 78 \] - **For \(n = 2\):** \[ a_2 = 78 - 6(1) = 72 \] - **Result:** **Correct** 3. **\(a_n = 84 - 6n\)** - **For \(n = 1\):** \[ a_1 = 84 - 6(1) = 78 \] - **For \(n = 2\):** \[ a_2 = 84 - 6(2) = 72 \] - **Result:** **Correct** 4. **\(a_n = -6n + 84\)** - This is algebraically equivalent to the third option and will yield the same results. - **Result:** **Correct** ### Conclusion: Only the first formula **does not** accurately describe the given arithmetic sequence. **Answer:** \(a_{n}=-6 n+90\) is not an accurate formula for the given sequence.