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The sequence given is \( a_n = \left\{ -2, 2, -\frac{8}{3}, 4, -\frac{32}{5}, \dots \right\} \), and we are tasked with finding \( a_{100} \).

### Step-by-Step Analysis:

1. **Pattern of Signs**: 
   The sequence alternates between negative and positive terms. This suggests a sign alternating factor of \( (-1)^n \).

2. **Numerical Pattern**:
   Observing the absolute values of the terms:
   - The first term is \( 2 \).
   - The second term is \( 2 \).
   - The third term is \( \frac{8}{3} \).
   - The fourth term is \( 4 \).
   - The fifth term is \( \frac{32}{5} \).

   The pattern in the numerators appears to be: \( 2, 2, 8, 4, 32 \).

   Comparing these terms, we see a doubling pattern emerging:
   - \( 2 = 2 \times 1 \)
   - \( 8 = 2^3 \)
   - \( 32 = 2^5 \)

   Therefore, the numerators seem to follow the pattern of \( 2^{n-1} \) for odd terms and \( 2^{n-2} \) for even terms.

3. **Denominators**:
   The denominators for odd-indexed terms appear to be \( 1, 3, 5, \dots \), forming the odd-number sequence.

4. **General Formula**:
   Based on this, we hypothesize the general term formula for \( a_n \) to be:
   \[
   a_n = (-1)^n \cdot \frac{2^{n-1}}{\text{odd number sequence corresponding to } n}
   \]

5. **Finding \( a_{100} \)**:
   To find \( a_{100} \), we need to consider whether the general pattern holds, and if so, calculate it accordingly.

I will now compute the exact value of \( a_{100} \) based on this analysis.The value of \( a_{100} \) in the given sequence is \( 562,949,953,421,312 \).

Would you like further details or clarifications on this result?

Here are five related questions for practice:
1. How would you describe the general form of an alternating sequence?
2. Can you deduce the 101st term using the pattern we identified?
3. What would change in the sequence if the numerators didn't follow powers of 2?
4. How does identifying a denominator pattern help solve sequences?
5. Can you explain why the sign alternates in this particular sequence?

**Tip**: When solving sequences, always check both the numerator and denominator patterns independently to avoid mistakes!

Given the sequence a_n = {-2, 2, -8/3, 4, -32/5, ...}, find a_100.

Explore how to find the 100th term (a_100) in a given alternating sequence. This problem covers key concepts in sequences, including alternating signs, patterns in numerators, and the odd-number sequence. Suitable for high school students in Grades 10-12.

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