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The given expression is:

\[
2 + 2(3) + 2(3^2) + 2(3^3) + \dots + 2(3^n)
\]

This is a geometric series where each term after the first is multiplied by 3. The first term is 2, and the common ratio \( r \) is 3.

The general formula for the sum \( S_n \) of a geometric series is:

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

Where:
- \( a \) is the first term (in this case, 2),
- \( r \) is the common ratio (in this case, 3),
- \( n \) is the highest exponent of 3.

However, since each term has a factor of 2, we can factor it out. So, the sum becomes:

\[
S_n = 2 \left( 1 + 3 + 3^2 + 3^3 + \dots + 3^n \right)
\]

This is a geometric series with the first term 1 and common ratio 3. Using the geometric sum formula for \( 1 + 3 + 3^2 + \dots + 3^n \), we have:

\[
S_n = 2 \cdot \frac{3^{n+1} - 1}{3 - 1}
\]

Simplifying:

\[
S_n = 2 \cdot \frac{3^{n+1} - 1}{2}
\]

So, the expression simplifies to:

\[
S_n = 3^{n+1} - 1
\]

Thus, the expression for the sum of the given series is:

\[
S_n = 3^{n+1} - 1
\]

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. What is the sum of the series when \( n = 4 \)?
2. How does the sum change if the common ratio is 4 instead of 3?
3. How would you modify the formula if the first term is not 2?
4. What is the general form of a geometric series?
5. How do geometric series differ from arithmetic series?

**Tip:** Always check the first few terms of a series to verify if it follows a geometric progression pattern before applying formulas.

Find an expression for 2 + 2(3) + 2(3^2) + ... + 2(3^n).

Learn how to find the sum of the geometric series 2 + 2(3) + 2(3^2) + ... + 2(3^n). This problem involves geometric series and algebraic concepts, suitable for Grades 10-12 students. It includes the step-by-step use of the geometric series sum formula.

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