Find the sum of the geometric series
27 + 9 + 3 +...+ 1/243. Express answer in simplest fractional form.

To find the sum of a geometric series, we use the formula:

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

where:
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the number of terms.

Given the series: 

\[
27 + 9 + 3 + \ldots + \frac{1}{243}
\]

### Step 1: Identify \(a\), \(r\), and \(n\)

1. **First term \(a\):** \(27\)
2. **Common ratio \(r\):** To find the common ratio, divide the second term by the first term:

\[
r = \frac{9}{27} = \frac{1}{3}
\]

3. **Number of terms \(n\):** We know that the last term is \(\frac{1}{243}\), and each term is multiplied by \(\frac{1}{3}\). To find \(n\), we use the formula for the \(n\)-th term of a geometric sequence:

\[
a \cdot r^{n-1} = \frac{1}{243}
\]

Substituting \(a = 27\) and \(r = \frac{1}{3}\):

\[
27 \cdot \left(\frac{1}{3}\right)^{n-1} = \frac{1}{243}
\]

Divide both sides by 27:

\[
\left(\frac{1}{3}\right)^{n-1} = \frac{1}{27 \cdot 243} = \frac{1}{6561}
\]

Since \(\left(\frac{1}{3}\right)^8 = \frac{1}{6561}\), we have \(n-1 = 8\), so \(n = 9\).

### Step 2: Calculate the Sum \(S_n\)

Now, use the formula for the sum of the first \(n\) terms:

\[
S_9 = 27 \frac{1 - \left(\frac{1}{3}\right)^9}{1 - \frac{1}{3}}
\]

Simplify:

\[
S_9 = 27 \frac{1 - \frac{1}{19683}}{\frac{2}{3}}
\]

\[
S_9 = 27 \times \frac{3}{2} \left(1 - \frac{1}{19683}\right)
\]

\[
S_9 = 27 \times \frac{3}{2} \times \frac{19682}{19683}
\]

\[
S_9 = \frac{81 \times 19682}{2 \times 19683}
\]

\[
S_9 = \frac{1594242}{39366}
\]

### Step 3: Simplify the Fraction

Both the numerator and the denominator are divisible by 3:

\[
S_9 = \frac{531414}{13122}
\]

This simplifies further to:

\[
S_9 = \frac{177138}{4374} = \frac{59046}{1458} = \frac{19682}{486}
\]

Further reducing gives:

\[
S_9 = \frac{3281}{81}
\]

### Conclusion

The sum of the geometric series is:

\[
\boxed{\frac{3281}{81}}
\]


Would you like more details on these steps or explanations? 

### Related Questions

1. What is the sum formula for an infinite geometric series?
2. How does the convergence of a geometric series depend on the common ratio?
3. Can you solve a similar problem where the common ratio is negative?
4. How do you determine the number of terms when the last term is given in geometric progressions?
5. What happens to the sum if the common ratio is equal to 1?

**Tip**: Always double-check your simplifications to avoid mistakes!

Find the sum of the geometric series 27 + 9 + 3 +...+ 1/243. Express the answer in simplest fractional form.

Discover how to find the sum of the geometric series 27 + 9 + 3 +...+ 1/243. Learn key algebraic concepts, including geometric series formulas and step-by-step problem-solving techniques. Suitable for students in Grades 9-12.

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