2. Determine the geometric sequence if the first term is $$ \frac{2 x}{5} $$ and the 4th term is $$ \frac{2 x^{4}}{5^{4}} $$.

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To determine the geometric sequence given the first term and the fourth term, follow these steps:

1. **Identify the Given Information:**
   - **First term ($$a_1$$)**: $$\frac{2x}{5}$$
   - **Fourth term ($$a_4$$)**: $$\frac{2x^4}{5^4}$$

2. **Understand the Geometric Sequence Formula:**
   - The $$n$$-th term of a geometric sequence is given by:
     $$
     a_n = a_1 \times r^{n-1}
     $$
     where $$r$$ is the common ratio.

3. **Find the Common Ratio ($$r$$):**
   - Using the fourth term:
     $$
     a_4 = a_1 \times r^3
     $$
     Plugging in the known values:
     $$
     \frac{2x^4}{5^4} = \frac{2x}{5} \times r^3
     $$
   - Solve for $$r^3$$:
     $$
     \frac{2x^4}{5^4} \div \frac{2x}{5} = r^3 \quad \Rightarrow \quad \frac{x^3}{5^3} = r^3
     $$
   - Taking the cube root:
     $$
     r = \frac{x}{5}
     $$

4. **Write the General Term of the Sequence:**
   - Substitute $$a_1$$ and $$r$$ into the general formula:
     $$
     a_n = \frac{2x}{5} \times \left(\frac{x}{5}\right)^{n-1}
     $$
   - Simplify:
     $$
     a_n = \frac{2x^n}{5^n}
     $$

5. **List the First Few Terms for Clarity:**
   - $$a_1 = \frac{2x}{5}$$
   - $$a_2 = \frac{2x^2}{5^2}$$
   - $$a_3 = \frac{2x^3}{5^3}$$
   - $$a_4 = \frac{2x^4}{5^4}$$
   - $$\vdots$$

**Conclusion:**
The geometric sequence is defined by the $$n$$-th term:
$$
a_n = \frac{2x^n}{5^n}
$$

**Final Answer:**
After simplifying, each term is 2 xⁿ divided by 5ⁿ. Thus, the geometric sequence is

aₙ = 2 xⁿ ⁄ 5ⁿ.
The geometric sequence is defined by the formula $$ a_n = \frac{2x^n}{5^n} $$.

2. Determine the geometric sequence if the first term is \( \frac{2 x}{5} \) and the 4th term is \( \frac{2 x^{4}}{5^{4}} \).

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