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?

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.