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To find the sum of a geometric series, there is a handy formula. A geometric series is a series with a constant ratio between successive terms. This specific formula makes it easy to sum all terms up to a certain number. The formula for the sum of the first \( n \) terms of a geometric series is:

• \( S_n = a \frac{1-r^n}{1-r} \)

Here is what each part means:

• \( S_n \): the sum of the series
• \( a \): the first term in the series
• \( r \): the common ratio between the terms
• \( n \): the number of terms

In our exercise, we used this formula to find the sum of the first six terms where the first term \( a = \left(\frac{2}{3}\right)^2 \) and the common ratio \( r = \frac{2}{3} \). By plugging these values into the formula, you can calculate the sum quickly and efficiently. It simplifies a potentially long arithmetic sum into a simple multiplication and division task, which saves a lot of time.

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. In the sequence, each term grows by the factor of this ratio.

• Example: A sequence like \( 2, 6, 18, 54 \) has a common ratio of 3.
• In our exercise, the sequence starts with \((\frac{2}{3})^2\) and has a common ratio of \( \frac{2}{3} \).

Understanding this term is crucial to solving problems involving geometric sequences. By identifying the first term and common ratio, you can write down any term in the sequence. Knowing this also helps in finding the sum using the sum of geometric series formula. The power of recognizing a sequence allows for predicting and managing patterns in numbers, making calculations much more accessible.

Evaluating a series is all about calculating the sum of its numbers. For a geometric series, this process becomes much straightforward using the geometric series formula.To evaluate a series, here’s what you usually do:

• Identify the first term \( a \).
• Find the common ratio \( r \).
• Decide how many terms \( n \) you want to sum up.
• Plug these values into the formula \( S_n = a \frac{1-r^n}{1-r} \) to get the result.

In the exercise you looked at, accurate evaluation meant substituting values correctly and performing arithmetic operations precisely. Understanding each part ensures you account for all variables, leading to the correct computation of the series sum. Moreover, practicing series evaluation strengthens your grasp on numerical patterns and expands problem-solving skills.