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:

= \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$$. Would you like more details, or do you have any questions? --- Here are some related questions: 1. What is the formula for the sum of an infinite geometric series? 2. How does the common ratio affect the convergence of a geometric series? 3. Can the sum of a geometric series be negative? 4. How do we handle geometric sequences with alternating signs? 5. How does the sum formula change for a geometric progression starting at a different index? **Tip:** For geometric series, if the absolute value of the common ratio $$|r| < 1$$, the series converges as $$n \to \infty$$.