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In a geometric series, the first term is crucial as it is the starting point for the sequence. It is often denoted by the letter \( a \). To find the first term in a given series, simply look at the first item in the sequence. In our problem, the expression starts with \((\frac{2}{3})^1\), which simplifies to \( \frac{2}{3} \). This value is our first term, \( a \). Understanding this concept provides the foundation for building the rest of the series, as each subsequent term will be a multiple of this first term, adjusted by the ratio that dictates the sequence's growth or reduction.

The common ratio in a geometric series determines how each term relates to the previous one. Understanding this ratio is essential as it indicates by how much each term increases or decreases. The common ratio, denoted by \( r \), is found by dividing any term in the sequence by the previous term. In our series, \( r = \frac{2}{3} \). This constant ratio repeats throughout the series, which distinguishes geometric sequences from other types. Knowing the common ratio allows us to predict the behavior of the entire series and compute subsequent terms effectively. Additionally, grasping this concept helps in determining whether a series converges or diverges, especially in infinite series.

The number of terms in a geometric series tells us how many elements are included in the sequence. This value is typically denoted by \( n \). In our example, the series contains six terms, which can be identified directly from the series notation \( \sum_{n=1}^{6} \). Knowing the exact number of terms is important because it influences the computation of the series' sum. It allows us to understand the sequence's limits and aids in applying the series sum formula accurately. Calculating a series without a clear understanding of the number of terms can lead to errors in determining its total value.

The sum of a geometric series is found using a specific formula. This formula helps to calculate the total of all terms in the series quickly. For a geometric series, the sum \( S \) of the first \( n \) terms is given by: \[ S = \frac{a(1 - r^n)}{1 - r} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

• Substitute in values: Using the example series, substitute the known values \( a = \frac{2}{3} \), \( r = \frac{2}{3} \), and \( n = 6 \).
• Calculate: Now, compute \( S = \frac{(\frac{2}{3})(1 - (\frac{2}{3})^6)}{1 - \frac{2}{3}} \).
• Result: After simplification, the sum \( S \) equals \( \frac{728}{243} \).

This formula is fundamental for efficiently computing sums of finite geometric series and offers a quick way to find the series' total value without manually adding each term.