\begin{tabular}{|l|l|}\hline Example 3: Find the \( n \)th Term \\ \begin{tabular}{|l|l|}\hline a. Write an equation for the nth term of the & b. Find the 9 th term of the sequence. \\ arithmetic sequence \( -12,-8,-4,0, \ldots . \). & \\ \hline\end{tabular} \\ \hline c. Which term of the sequence is 32 ? & \\ \hline\end{tabular}

Sure, let's tackle **Example 3** step by step. ### Given: An arithmetic sequence: \(-12, -8, -4, 0, \ldots\) ### a. Write an equation for the \( n \)th term of the arithmetic sequence. **Solution:** An arithmetic sequence has the form: \[ a_n = a_1 + (n - 1)d \] where: - \( a_n \) is the \( n \)th term, - \( a_1 \) is the first term, - \( d \) is the common difference, - \( n \) is the term number. **Identify the values:** - First term (\( a_1 \)): \(-12\) - To find the common difference (\( d \)): \[ d = a_2 - a_1 = (-8) - (-12) = 4 \] **Write the equation:** \[ a_n = -12 + (n - 1) \times 4 \] \[ a_n = -12 + 4n - 4 \] \[ a_n = 4n - 16 \] **Equation for the \( n \)th term:** \[ a_n = 4n - 16 \] ### b. Find the 9th term of the sequence. **Solution:** Use the equation derived in part (a): \[ a_n = 4n - 16 \] **Find \( a_9 \):** \[ a_9 = 4(9) - 16 \] \[ a_9 = 36 - 16 \] \[ a_9 = 20 \] **The 9th term is:** \( 20 \) ### c. Which term of the sequence is 32? **Solution:** We need to find \( n \) such that \( a_n = 32 \). Using the equation: \[ a_n = 4n - 16 \] \[ 32 = 4n - 16 \] \[ 32 + 16 = 4n \] \[ 48 = 4n \] \[ n = \frac{48}{4} \] \[ n = 12 \] **The term 32 is the 12th term of the sequence.**