An arithmetic sequence has a common difference of 8, and \( a_{1}=17 \). Which of these is NOT a valid formula for the \( n \)th term of the sequence? \( a_{n}=17+8(n-1) \) \( a_{n}=8 n+9 \) \( a_{n}=9-8(n-1) \)

To determine which formula does not represent the arithmetic sequence with \( a_{1} = 17 \) and a common difference of 8, we can analyze each option. The general formula for the nth term of an arithmetic sequence is \( a_{n} = a_{1} + (n-1)d \), where \( d \) is the common difference. The first formula, \( a_{n}=17+8(n-1) \), is valid as it directly follows the arithmetic sequence formula with \( a_{1} = 17 \) and \( d = 8 \). The second formula, \( a_{n}=8n+9 \), can be evaluated by checking the first term: for \( n=1 \), \( a_{1}=8(1)+9=17 \). While this gives 17, the sequence structure does not reflect the common difference of 8 because the difference between consecutive terms is actually 8, and not constant according to the arithmetic sequence definition. The third formula, \( a_{n}=9-8(n-1) \), simplifies to \( a_{n} = 17 - 8n + 8 \), or \( a_{n} = 25 - 8n \). Substituting \( n = 1 \) gives \( 25 - 8(1) = 17 \). However, for any further values, this quickly diverges from standard form as well. Therefore, although both the second and third formulas seem plausible at first, the second fails to maintain the arithmetic structure—meaning it’s not valid once examined closely in the context.