 Determine the summation notation for the series 3 + 9 + 27+81+243

The given series is:

\[ 3 + 9 + 27 + 81 + 243 \]

This is a geometric series where the first term \( a = 3 \) and the common ratio \( r = 3 \), because each term is multiplied by 3 to get the next term.

### General Formula for a Geometric Series
A geometric series can be written as:

\[
S_n = a + ar + ar^2 + \dots + ar^{n-1}
\]

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

### For the given series:
- \( a = 3 \),
- \( r = 3 \),
- Number of terms \( n = 5 \).

Thus, the summation notation for the given series is:

\[
\sum_{k=0}^{4} 3 \cdot 3^k
\]

Or equivalently:

\[
\sum_{k=0}^{4} 3^{k+1}
\]

This represents the sum of the terms \( 3 + 9 + 27 + 81 + 243 \).

Would you like any further details or clarifications?

Here are five related questions you could explore:
1. What is the sum of this geometric series?
2. How does the formula change for an infinite geometric series?
3. What happens if the common ratio is less than 1?
4. Can you generalize this series for any number of terms?
5. How do you find the sum of a finite geometric series?

**Tip:** In geometric series, the sum can be found easily using the formula \( S_n = \frac{a(1 - r^n)}{1 - r} \) if \( |r| \neq 1 \).

Determine the summation notation for the series 3 + 9 + 27 + 81 + 243

Learn how to express the series 3 + 9 + 27 + 81 + 243 in summation notation. This problem explores the concept of geometric series, using the first term and common ratio to form a general formula. Ideal for students in Grades 9-12.

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