Write an explicit formula for a, start subscript, n, end subscript*an*​, the n, start superscript, th, end superscript*n*th term of the sequence 32,−16,8

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.

Learn how to write an explicit formula for the nth term of a geometric sequence, specifically analyzing the sequence 32, -16, 8. Understand the concepts of geometric sequences, common ratios, and the formula a_n = 32 * (-1/2)^(n-1). Suitable for students in Grades 10-12.

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