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

To find the \( n \)th term of the geometric sequence \( 3, -12, 48, -192, \ldots \), we start by identifying the first term and the common ratio. The first term \( a \) is \( 3 \), and the common ratio \( r \) can be found by dividing the second term by the first term: \[ r = \frac{-12}{3} = -4. \] We can describe the \( n \)th term of a geometric sequence with the formula \( a_n = a \cdot r^{(n-1)} \). Plugging in our values, the equation for the \( n \)th term is: \[ a_n = 3 \cdot (-4)^{(n-1)}. \] So, for this sequence, each term is obtained by multiplying the previous term by \(-4\), resulting in the formula for the \( n \)th term!