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

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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:**
$$
a_n = 3 \cdot (-4)^{\,n-1}
$$
The $$ n $$th term of the sequence is $$ a_n = 3 \times (-4)^{n-1} $$.

Write an equation for the th term of the geometric sequence
.

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Write an equation for the th term of the geometric sequence .