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?

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.