The problem asks to find the sum of a geometric series, starting with the terms:

$3 + \frac{9}{2} + \cdots + \frac{6561}{128}$

Step-by-step approach:

1. Identify the first term ($a$): The first term, $a$, is clearly $3$.

2. Identify the common ratio ($r$): The second term is $\frac{9}{2}$. The ratio between the second term and the first term is: $r = \frac{\frac{9}{2}}{3} = \frac{9}{2} \times \frac{1}{3} = \frac{3}{2}$

   So, the common ratio is $r = \frac{3}{2}$.

3. Determine the number of terms ($n$): The series goes up to $\frac{6561}{128}$, so we need to check how many terms this corresponds to. The general formula for the $n$-th term in a geometric series is: $a_n = a \times r^{n-1}$ Set this equal to $\frac{6561}{128}$ and solve for $n$: $\frac{6561}{128} = 3 \times \left(\frac{3}{2}\right)^{n-1}$ Dividing both sides by 3: $\frac{6561}{128 \times 3} = \left(\frac{3}{2}\right)^{n-1}$ $\frac{6561}{384} = \left(\frac{3}{2}\right)^{n-1}$ Simplifying $\frac{6561}{384}$: $\frac{6561}{384} = 17.09$ Now, solve for $n$ using logarithms: $\log \left( 17.09 \right) = (n-1) \log \left( \frac{3}{2} \right)$

4. Sum of the geometric series: The formula for the sum of a geometric series is: $S_n = \frac{a(1 - r^n)}{1 - r}$ After determining $n$, substitute into this formula to find the sum.

Let me calculate the exact value for you.The number of terms in the series is $n = 7$ (since $n$ rounds to 7). The sum of the geometric series is approximately $96.516$ when rounded to 3 significant figures.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How is the formula for the sum of a finite geometric series derived?
2. What happens if the common ratio of a geometric series is less than 1?
3. Can you calculate the sum of an infinite geometric series with $r = \frac{3}{2}$?
4. How do we deal with geometric series where the common ratio is negative?
5. What is the difference between arithmetic and geometric series in terms of growth?

Tip: Always check the common ratio carefully when working with geometric series. It determines how fast or slow the series grows!