Write an equation for the \( n \)th term of the geometric sequence \( 3,-12,48,-192, \ldots \).

To find the equation for the \( n \)th term of the geometric sequence \( 3, -12, 48, -192, \ldots \), follow these steps: 1. **Identify the first term (\( a_1 \)) and the common ratio (\( r \)):** - The first term \( a_1 = 3 \). - The common ratio \( r \) is found by dividing the second term by the first term: \[ r = \frac{-12}{3} = -4 \] - To confirm, check the ratio between subsequent terms: \[ \frac{48}{-12} = -4 \quad \text{and} \quad \frac{-192}{48} = -4 \] The common ratio \( r = -4 \) is consistent. 2. **Use the formula for the \( n \)th term of a geometric sequence:** \[ a_n = a_1 \cdot r^{n-1} \] 3. **Substitute the known values into the formula:** \[ a_n = 3 \cdot (-4)^{n-1} \] **Final Equation:** \[ a_n = 3 \cdot (-4)^{\,n-1} \]