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The problem asks for the summation of the first eight terms of a sequence. The sequence provided is \(3, \_, 27, 81, \dots\).

First, let's identify the sequence type and determine the missing term and the pattern.

### Step 1: Identify the Sequence Type
The sequence appears to be geometric because the ratio between successive terms is constant.

Given:
- First term (\(a_1\)) = 3
- Third term (\(a_3\)) = 27
- Fourth term (\(a_4\)) = 81

### Step 2: Determine the Common Ratio (\(r\))
The common ratio \(r\) can be found using the third and first terms:
\[
r = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \frac{27}{a_2} \quad \text{and} \quad \frac{81}{27} = r
\]
Since \(81 \div 27 = 3\), we can deduce that \(r = 3\).

Using the ratio to find \(a_2\):
\[
a_2 = \frac{27}{3} = 9
\]

### Step 3: Verify the Sequence
Now, the sequence is:
\[
3, 9, 27, 81, \dots
\]
This confirms the common ratio is indeed \(3\).

### Step 4: Summation of the First 8 Terms
The sum \(S_n\) of the first \(n\) terms of a geometric sequence can be calculated using:
\[
S_n = a_1 \times \frac{r^n - 1}{r - 1}
\]
For \(n = 8\), \(a_1 = 3\), and \(r = 3\):
\[
S_8 = 3 \times \frac{3^8 - 1}{3 - 1}
\]
Calculate:
\[
3^8 = 6561
\]
Thus:
\[
S_8 = 3 \times \frac{6561 - 1}{2} = 3 \times \frac{6560}{2} = 3 \times 3280 = 9840
\]

### Answer:
The summation of the first eight terms is **9840**.

Would you like more details on any part of this solution?

Here are 5 related questions you could explore:
1. How do you derive the formula for the sum of a geometric series?
2. What is the sum of the first 10 terms of the same sequence?
3. How does the common ratio affect the growth of a geometric sequence?
4. What if the common ratio were a fraction—how would that impact the sequence?
5. Can you explain how to find the sum if the sequence were infinite?

**Tip:** Always verify the common ratio by checking multiple terms to ensure the sequence is consistent.

From the given sequence: 3, _, 27, 81, find the summation up to the 8th term.

Find the summation of the first 8 terms of the geometric sequence 3, _, 27, 81. The missing term is calculated, and the sum is determined using the geometric series formula. Suitable for high school students learning about geometric progressions.

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