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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!

Find the sum of the geometric series 3 + 9/2 + ... + 6561/128, correct to 3 significant figures.

This page provides a detailed solution to finding the sum of the geometric series starting with 3 + 9/2 + ... up to 6561/128. Learn how to apply the geometric series sum formula and solve for the total sum correct to 3 significant figures. Ideal for students in Grades 10-12.

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