Determine the formula for the "n^th" term of the geometric sequence below.
2,-16, 18. -54, 162...

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.

Discover the formula for the nth term of a geometric sequence with a step-by-step explanation. Learn how to apply this formula to find specific terms in the sequence. Suitable for high school students studying geometric sequences.

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