To write an explicit formula for the $n$th term ($a_n$) of the sequence $32, -16, 8, \ldots$, we first need to determine the type of sequence and identify its pattern.

Step-by-Step Solution

1. Identify the type of sequence:

   Let's look at the sequence $32, -16, 8, \ldots$. To determine if it's arithmetic or geometric, we calculate the ratio between consecutive terms.

   • First, calculate the ratio between the second term and the first term:

   $\frac{-16}{32} = -\frac{1}{2}$

   • Next, calculate the ratio between the third term and the second term:

   $\frac{8}{-16} = -\frac{1}{2}$

   Since the ratio between consecutive terms is constant $(- \frac{1}{2})$, this is a geometric sequence.

2. Identify the first term and the common ratio:

   • The first term $a_1$ of the sequence is $32$.
   • The common ratio $r$ is $-\frac{1}{2}$.

3. Use the formula for the $n$th term of a geometric sequence:

   The formula for the $n$th term of a geometric sequence is:

   $a_n = a_1 \cdot r^{n-1}$

   where:

   • $a_1 = 32$ (the first term)
   • $r = -\frac{1}{2}$ (the common ratio)

4. Substitute the values into the formula:

   $a_n = 32 \cdot \left(-\frac{1}{2}\right)^{n-1}$

Final Explicit Formula

The explicit formula for the $n$th term of the sequence $32, -16, 8, \ldots$ is:

$a_n = 32 \cdot \left(-\frac{1}{2}\right)^{n-1}$

This formula can be used to find any term in the sequence by substituting the value of $n$.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would you calculate the 5th term of this sequence using the formula $a_n = 32 \cdot \left(-\frac{1}{2}\right)^{n-1}$?
2. What is the formula for the sum of the first $n$ terms of a geometric sequence?
3. How do you determine if a sequence is arithmetic or geometric just by looking at a few terms?
4. What happens to the explicit formula if the common ratio changes to $\frac{1}{3}$?
5. How do you find the common ratio of a geometric sequence if only two terms are given?

Tip: Always check if the common ratio or common difference is consistent to correctly identify the type of sequence.