ERROR ANALYSIS Describe and correct the error in writing a rule for the \( n \)th term of the arithmetic sequence \( 22,9,-4,-17 \), \( -30, \ldots \). The first term is 22 and the common difference Is -13 . \( a_{n}=-13+(n-1)(22) \) \( a_{n}=-35+22 n \) \( \begin{array}{l}\text { The formula is incorrectly calculated. } \\ \text { The first term and common difference were switched. } \\ a_{n}=\square \text { The formula should be } a_{n}=a_{1}(n+1)+d \text {. Tormula should be } a_{n}=a_{1}(d)+(n-1) \text {. }\end{array} \)

Let's analyze the error in writing the rule for the \( n \)th term of the arithmetic sequence \( 22, 9, -4, -17, -30, \ldots \). ### Known Conditions: 1. The first term \( a_1 = 22 \). 2. The common difference \( d = 9 - 22 = -13 \). ### Correct Formula for the \( n \)th Term: The general formula for the \( n \)th term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1) \cdot d \] ### Step-by-Step Correction: 1. **Identify the error in the original formula:** - The original formula provided is \( a_n = -13 + (n - 1)(22) \). - Here, the common difference \( d \) and the first term \( a_1 \) are incorrectly placed. 2. **Substituting the correct values:** - The correct formula should use \( a_1 = 22 \) and \( d = -13 \): \[ a_n = 22 + (n - 1)(-13) \] 3. **Simplifying the formula:** - Distributing the common difference: \[ a_n = 22 - 13(n - 1) \] - This simplifies to: \[ a_n = 22 - 13n + 13 \] - Combining like terms: \[ a_n = 35 - 13n \] ### Final Correct Formula: Thus, the correct formula for the \( n \)th term of the arithmetic sequence is: \[ a_n = 35 - 13n \] ### Summary of Errors: - The first term and common difference were switched in the original formula. - The correct formula is \( a_n = a_1 + (n - 1) \cdot d \), which leads to \( a_n = 35 - 13n \) after proper substitution and simplification.