Write each of the following expressions using summation notation.
(a) 1
3 + 1
9 + 1
27 + 1
81 + 1
243 + 1
729

We can observe that the terms in the given expression \( 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \frac{1}{243}, \frac{1}{729} \) follow a geometric progression. The general term of this geometric series is \( \frac{1}{3^n} \), where \( n \) starts at 0 and increases.

Thus, the given expression can be written in summation notation as:

\[
\sum_{n=0}^{6} \frac{1}{3^n}
\]

This represents the sum of the terms \( \frac{1}{3^n} \) for \( n \) ranging from 0 to 6.

Do you need further details or have any questions? Here are 5 questions that could expand your understanding:

1. How do you determine the common ratio in a geometric series?
2. Can you derive the sum formula for an infinite geometric series?
3. What is the difference between finite and infinite series?
4. How would the summation notation change if the common ratio were different?
5. How can you compute the sum of the series without using summation notation?

**Tip:** In a geometric series, the ratio between consecutive terms is constant. This makes it easier to recognize and express the series using summation notation.

Write each of the following expressions using summation notation. (a) 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729

Learn how to express the geometric series 1/3 + 1/9 + 1/27 + 1/81 + 1/243 + 1/729 using summation notation. This problem covers key concepts like geometric progressions and summation, suitable for students in Grades 9-11.

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