To determine the formula for the $n^{th}$ term of the given geometric sequence, follow these steps:

1. Identify the first term $a_1$:

   • The first term $a_1$ is $2$.

2. Determine the common ratio $r$:

   • The common ratio $r$ is found by dividing any term by the previous term. For example: $r = \frac{-16}{2} = -8$ So, the common ratio $r = -8$.

3. Write the general formula for the $n^{th}$ term of a geometric sequence:

   • The formula for the $n^{th}$ term of a geometric sequence is given by: $a_n = a_1 \cdot r^{n-1}$

4. Substitute the values of $a_1$ and $r$:

   • Here, $a_1 = 2$ and $r = -8$, so the formula becomes: $a_n = 2 \cdot (-8)^{n-1}$

Thus, the formula for the $n^{th}$ term of the sequence is: $\boxed{a_n = 2 \cdot (-8)^{n-1}}$

Would you like details on how to apply this formula, or do you have any other questions?

Here are some related questions:

1. What is the 5th term of the sequence using the derived formula?
2. How does the common ratio affect the growth or decay of a geometric sequence?
3. Can the formula for the $n^{th}$ term be used to find the sum of the first $n$ terms?
4. What happens if the common ratio $r$ is a fraction between 0 and 1?
5. How can you determine if a sequence is geometric just by looking at it?

Tip: In geometric sequences, the common ratio can be negative, leading to an alternating sign pattern in the sequence.