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To find the general term \( a_n \) of the sequence, let's examine the pattern of the terms:

The terms are:
\[
-2, \quad \frac{4}{3}, \quad -\frac{8}{9}, \quad \frac{16}{27}, \quad -\frac{32}{81}, \dots
\]

### Step 1: Look for a Pattern in the Numerators and Denominators
1. **Numerator Pattern**: The numerators are \( 2, 4, 8, 16, 32, \dots \), which follow a pattern of powers of 2:
   \[
   2 = 2^1, \quad 4 = 2^2, \quad 8 = 2^3, \quad 16 = 2^4, \quad 32 = 2^5, \dots
   \]
   So, the \( n \)-th term's numerator is \( 2^n \).

2. **Denominator Pattern**: The denominators are \( 1, 3, 9, 27, 81, \dots \), which are powers of 3:
   \[
   1 = 3^0, \quad 3 = 3^1, \quad 9 = 3^2, \quad 27 = 3^3, \quad 81 = 3^4, \dots
   \]
   The \( n \)-th term's denominator is \( 3^{n-1} \).

### Step 2: Consider the Sign Pattern
The sequence alternates in sign: \( -, +, -, +, - \dots \). This can be represented by \( (-1)^{n+1} \) since it starts negative when \( n = 1 \).

### Step 3: Form the General Formula
Combining these observations, we can write the general term \( a_n \) as:
\[
a_n = (-1)^{n+1} \cdot \frac{2^n}{3^{n-1}}
\]

### Final Answer
\[
a_n = (-1)^{n+1} \frac{2^n}{3^{n-1}}
\]

Would you like a detailed explanation of each step? 

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#### Related Questions to Explore
1. How can we identify the general term of a sequence based on alternating signs?
2. What other sequences have numerators and denominators as powers of different integers?
3. How does the exponent in the general term change if the sequence starts with a positive number?
4. Can this approach be applied to sequences where terms decrease in magnitude?
5. How would the formula change if the sequence started with \( n = 0 \)?

#### Tip:
In sequences with alternating signs, using \( (-1)^n \) or \( (-1)^{n+1} \) helps control the sign pattern effectively.

Find a formula for the general term a_n of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) Sequence: -2, 4/3, -8/9, 16/27, -32/81...

Learn how to find the general term of the sequence -2, 4/3, -8/9, 16/27, -32/81... This problem covers key algebraic concepts in sequences, including alternating signs and powers of integers. Suitable for students in Grades 9-12.

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