Evaluating Infinite Geometric Series S_5 = 600 + 300 + 150 + ...

In the realm of mathematics, the concept of infinite series holds a significant position. Among these, geometric series, where each term is multiplied by a constant ratio to obtain the next, are particularly intriguing. This article delves into the evaluation of an infinite geometric series, specifically focusing on the series S_5 represented by 600 + 300 + 150 + .... We will explore the underlying principles, apply the relevant formulas, and arrive at the correct answer from the given options. Understanding geometric series is crucial in various fields, including calculus, physics, and finance, making this a valuable exploration.

Before diving into the specifics of the given series, let's establish a firm grasp on the fundamentals of geometric series. A geometric series is a sequence of numbers where each term is obtained by multiplying the preceding term by a constant factor, known as the common ratio. This common ratio, often denoted by 'r', is the cornerstone of geometric series. The general form of a geometric series is represented as:

a, ar, ar^2, ar^3, ...

where 'a' is the first term and 'r' is the common ratio. The behavior of a geometric series, particularly its convergence or divergence, hinges on the value of 'r'. If the absolute value of 'r' is less than 1 (|r| < 1), the series converges, meaning that the sum of its infinite terms approaches a finite value. Conversely, if |r| ≥ 1, the series diverges, and the sum grows infinitely large.

The sum of an infinite geometric series, when it converges, can be calculated using a specific formula. This formula is a powerful tool for evaluating series like the one we are addressing in this article. The formula for the sum (S) of an infinite geometric series is:

S = a / (1 - r)

where 'a' is the first term and 'r' is the common ratio, with the condition that |r| < 1. This formula stems from the limit of the partial sums of the series as the number of terms approaches infinity. It provides a direct method to find the sum, making the evaluation process efficient and accurate.

Now, let's apply our understanding of geometric series to the given series: S_5 = 600 + 300 + 150 + .... The first step in evaluating this series is to identify its key parameters: the first term ('a') and the common ratio ('r'). These parameters are essential for applying the sum formula and determining the series' convergence. In this series, the first term, 'a', is clearly 600. This is the starting point of the sequence, and it forms the basis for all subsequent terms.

To find the common ratio, 'r', we need to examine the relationship between consecutive terms. The common ratio is the factor by which each term is multiplied to obtain the next. We can calculate 'r' by dividing any term by its preceding term. For instance, dividing the second term (300) by the first term (600) gives us:

r = 300 / 600 = 1/2 = 0.5

Similarly, dividing the third term (150) by the second term (300) also yields 0.5, confirming that the common ratio is indeed 0.5. This consistency is a hallmark of geometric series, where the ratio between consecutive terms remains constant throughout the sequence. The value of 'r' is crucial because it dictates whether the series converges and, if so, what its sum will be. With 'a' = 600 and 'r' = 0.5, we have the necessary components to proceed with evaluating the sum of the series.

With the first term ('a') and the common ratio ('r') identified, we can now proceed to evaluate the sum of the infinite geometric series S_5. Recall that 'a' = 600 and 'r' = 0.5. The formula for the sum of an infinite geometric series is:

S = a / (1 - r)

This formula is applicable when the absolute value of the common ratio is less than 1, which is the case here as |0.5| < 1. This condition ensures that the series converges, and the sum approaches a finite value. Substituting the values of 'a' and 'r' into the formula, we get:

S = 600 / (1 - 0.5)

Simplifying the denominator, we have:

S = 600 / 0.5

Now, dividing 600 by 0.5, we obtain:

S = 1200

Therefore, the sum of the infinite geometric series S_5 = 600 + 300 + 150 + ... is 1200. This result signifies that as we continue adding terms in this series, the sum will get closer and closer to 1200, never exceeding it. The convergence of the series is a direct consequence of the common ratio being less than 1, allowing us to use the formula and arrive at a definitive sum. The evaluation of S_5 demonstrates the power of the geometric series formula in handling infinite sums.

Having calculated the sum of the series S_5 to be 1200, we now need to compare this result with the given answer choices. The options provided are:

A. 1,162.5 B. 581.25 C. 37.5 D. 18,600

By comparing our calculated sum (1200) with these options, we can clearly see that none of the given choices match the correct answer. This discrepancy could be due to an error in the provided options or a misunderstanding of the question's context. However, based on our calculations and the principles of geometric series, the sum of the series S_5 = 600 + 300 + 150 + ... is definitively 1200.

It is essential in mathematical problem-solving to verify the results and ensure they align with the given options or the expected outcome. In this case, the absence of 1200 as an option highlights the importance of double-checking both the calculations and the provided choices. While the mathematical process leads us to 1200, the practical aspect of selecting the correct answer requires a match between the calculated result and the available options. In situations like this, it's prudent to re-examine the problem statement and the steps taken to arrive at the solution, ensuring no errors were made in the process.

In this article, we embarked on a journey to evaluate the infinite geometric series S_5 = 600 + 300 + 150 + .... We began by establishing a solid understanding of geometric series, their properties, and the conditions for convergence. We identified the first term ('a') as 600 and the common ratio ('r') as 0.5. Applying the formula for the sum of an infinite geometric series, S = a / (1 - r), we calculated the sum to be 1200. This result signifies the value that the series approaches as we add an infinite number of terms.

However, when comparing our calculated sum with the provided answer choices, we found no match. This highlights a crucial aspect of problem-solving: the need for verification and alignment with the given options. While our mathematical process yielded 1200 as the sum, the absence of this value among the choices suggests a potential discrepancy in the options themselves or a need to re-evaluate the problem's context.

The exploration of S_5 provides valuable insights into the behavior of infinite geometric series and the application of the sum formula. It underscores the importance of accurate calculations and the critical step of comparing results with available choices. Understanding geometric series is fundamental in mathematics and has wide-ranging applications in various fields. This exercise reinforces the principles involved and enhances our problem-solving skills in the realm of infinite series.

In summary, the evaluation of S_5 demonstrates the power of mathematical tools in handling infinite sums, while also emphasizing the importance of verification and critical analysis in the problem-solving process. The calculated sum of 1200, though not present in the given options, stands as the correct answer based on the principles of geometric series.