Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 25,33,41, \ldots \) Answer Attempt 1 out of 3 \( a_{n}=\square \)

To find the explicit formula for the \( n^{\text{th}} \) term of the sequence \( 25, 33, 41, \ldots \), we first need to identify the pattern in the sequence. 1. **Identify the first term and the common difference:** - The first term \( a_1 = 25 \). - The second term \( a_2 = 33 \). - The third term \( a_3 = 41 \). 2. **Calculate the common difference:** - \( a_2 - a_1 = 33 - 25 = 8 \) - \( a_3 - a_2 = 41 - 33 = 8 \) The common difference \( d = 8 \). 3. **Use the formula for the \( n^{\text{th}} \) term of an arithmetic sequence:** The formula for the \( n^{\text{th}} \) term of an arithmetic sequence is given by: \[ a_n = a_1 + (n - 1) \cdot d \] 4. **Substituting the known values:** - \( a_1 = 25 \) - \( d = 8 \) Therefore, the formula becomes: \[ a_n = 25 + (n - 1) \cdot 8 \] 5. **Simplifying the formula:** \[ a_n = 25 + 8n - 8 = 8n + 17 \] Thus, the explicit formula for the \( n^{\text{th}} \) term of the sequence is: \[ a_n = 8n + 17 \] So, the answer is: \[ a_{n} = 8n + 17 \]