Listen ERROR ANALYSIS Describe and correct the error in writing a rule for the \( n \)th term of the arithmetic sequence. \[ 22,9,-4,-17,-30, \ldots . \] Use \( a_{1}=22 \) and \( d=-13 \). \[ \begin{array}{l} a_{n}=a_{1}+n d \\ a_{n}=22+n(-13) \\ a_{n}=22-13 n \end{array} \] The formula should be \( a_{n}=a_{1}(n-1) \). The formula should be \( a_{n}=a_{1}(n-1) d \). The formula should be \( a_{n}=d+n a_{1} \). The formula should be \( a_{n}=a_{1}+(n-1) d \). \[ a_{n}=\square \]

Nice work analyzing the sequence! However, the error lies in the initial formula used to derive the \( n \)th term for an arithmetic sequence. The correct formula is \( a_{n} = a_{1} + (n-1)d \), not just \( a_{1} + nd \). This small adjustment accounts for the fact that the first term doesn’t have a full cycle of the common difference \( d \). Now for the correction: substituting \( a_{1} = 22 \) and \( d = -13 \) into the corrected formula gives you \( a_{n} = 22 + (n-1)(-13) \). Simplifying that would provide the correct expression for the \( n \)th term as \( a_{n} = 22 - 13(n - 1) \). Your endeavors will surely pay off with this adjustment!