Find the sum of this geometric sequence: 2 + 6/7 + 18/49 +... + 2(3/7)^8

We are tasked with finding the sum of the geometric sequence:

\[
2 + \frac{6}{7} + \frac{18}{49} + \dots + 2\left(\frac{3}{7}\right)^8
\]

This sequence is a geometric progression (GP) with:

- First term \( a = 2 \)
- Common ratio \( r = \frac{3}{7} \)

The sum of the first \( n \) terms of a geometric sequence is given by the formula:

\[
S_n = a \frac{1 - r^n}{1 - r}
\]

### Step 1: Identify the relevant values
- The first term \( a = 2 \)
- The common ratio \( r = \frac{3}{7} \)
- The number of terms is \( n = 9 \) (since we go from the first term to \( 2\left(\frac{3}{7}\right)^8 \), this represents 9 terms).

### Step 2: Plug values into the formula
Now substitute \( a = 2 \), \( r = \frac{3}{7} \), and \( n = 9 \) into the sum formula:

\[
S_9 = 2 \frac{1 - \left(\frac{3}{7}\right)^9}{1 - \frac{3}{7}}
\]

### Step 3: Simplify the expression
First, simplify the denominator:

\[
1 - \frac{3}{7} = \frac{7 - 3}{7} = \frac{4}{7}
\]

Thus, the sum becomes:

\[
S_9 = 2 \times \frac{1 - \left(\frac{3}{7}\right)^9}{\frac{4}{7}}
\]

This simplifies to:

\[
S_9 = 2 \times \frac{7}{4} \times \left(1 - \left(\frac{3}{7}\right)^9\right)
\]

Now let's compute \( \left(\frac{3}{7}\right)^9 \).

\[
\left(\frac{3}{7}\right)^9 = \frac{3^9}{7^9}
\]

I'll calculate this next.There was a technical issue calculating \( \left(\frac{3}{7}\right)^9 \). Let me compute it directly for you:

\[
3^9 = 19683
\]
\[
7^9 = 40353607
\]

Thus, \( \left(\frac{3}{7}\right)^9 = \frac{19683}{40353607} \).

### Step 4: Substitute back into the sum expression

We now substitute this value back into the sum formula:

\[
S_9 = 2 \times \frac{7}{4} \times \left(1 - \frac{19683}{40353607}\right)
\]

Simplifying the term inside the parentheses:

\[
1 - \frac{19683}{40353607} = \frac{40353607 - 19683}{40353607} = \frac{40333924}{40353607}
\]

Now, substitute this into the sum:

\[
S_9 = 2 \times \frac{7}{4} \times \frac{40333924}{40353607}
\]

Simplifying further:

\[
S_9 = \frac{7 \times 40333924}{2 \times 4 \times 40353607}
= \frac{282337468}{322828856}
\]

Finally, reduce the fraction if possible:

\[
S_9 \approx 0.8748
\]

### Final Answer:
The sum of the geometric sequence is approximately \( S_9 \approx 0.8748 \).

Would you like more details, or do you have any questions?

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Here are some related questions:

1. What is the formula for the sum of an infinite geometric series?
2. How does the common ratio affect the convergence of a geometric series?
3. Can the sum of a geometric series be negative?
4. How do we handle geometric sequences with alternating signs?
5. How does the sum formula change for a geometric progression starting at a different index?

**Tip:** For geometric series, if the absolute value of the common ratio \( |r| < 1 \), the series converges as \( n \to \infty \).

Discover how to find the sum of the geometric sequence 2 + 6/7 + 18/49 +... + 2(3/7)^8 using the geometric series sum formula. This problem involves key concepts of geometric progressions and summation techniques, suitable for high school students (Grades 10-12).

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