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To solve for \( a_7 \) in the given sequence, let's analyze the pattern in the terms provided:

The sequence is given as:
\[
\left\{ a_1, a_2, a_3, a_4, \dots \right\} = \left\{ \frac{10}{3}, \frac{20}{9}, \frac{40}{27}, \frac{80}{81}, \dots \right\}
\]

### Step 1: Identify the Pattern
Looking at the terms:
- \( a_1 = \frac{10}{3} \)
- \( a_2 = \frac{20}{9} \)
- \( a_3 = \frac{40}{27} \)
- \( a_4 = \frac{80}{81} \)

Notice that both the numerators and denominators appear to follow specific patterns:
- The numerators: \(10, 20, 40, 80, \dots\) seem to double each time. This is a geometric progression with a common ratio of \(2\).
- The denominators: \(3, 9, 27, 81, \dots\) are powers of \(3\), specifically \(3^1, 3^2, 3^3, 3^4, \dots\).

### Step 2: General Formula for \(a_n\)
From the patterns observed:
1. The numerator for \( a_n \) is \(10 \times 2^{n-1}\).
2. The denominator for \( a_n \) is \(3^n\).

Thus, the general term for the sequence \( a_n \) can be written as:
\[
a_n = \frac{10 \cdot 2^{n-1}}{3^n}
\]

### Step 3: Calculate \( a_7 \)
To find \( a_7 \), substitute \( n = 7 \) into the formula:
\[
a_7 = \frac{10 \cdot 2^{7-1}}{3^7} = \frac{10 \cdot 2^6}{3^7}
\]

Now, calculate the powers:
- \( 2^6 = 64 \)
- \( 3^7 = 2187 \)

Therefore:
\[
a_7 = \frac{10 \cdot 64}{2187} = \frac{640}{2187}
\]

### Answer
\[
a_7 = \frac{640}{2187}
\]

Would you like further details on any steps, or do you have any questions?

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Here are some related questions to expand your understanding:
1. How would you find \( a_{10} \) in this sequence?
2. What is the general rule to identify the \( n \)-th term in a sequence?
3. Can you identify if this sequence converges or diverges?
4. What are some examples of other geometric sequences with different common ratios?
5. How can you determine if a sequence is arithmetic or geometric?

**Tip:** Recognizing the patterns in sequences is essential for quickly finding general formulas for \( n \)-th terms. Look for common ratios or differences between terms.

Dada la sucesión {a1, a2, a3, a4, ...} = {10/3, 20/9, 40/27, 80/81, ...}, determine a7.

Learn how to find the 7th term (a7) in the sequence {10/3, 20/9, 40/27, 80/81, ...}. This problem involves geometric sequences, identifying patterns in numerators and denominators, and applying the general formula for geometric sequences. Suitable for students in Grades 10-12.

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