Question

Find the sum of the series \(2 \sqrt{3}, 2 \sqrt{2}, \frac{4}{\sqrt{3}}, \ldots\) (a) \(6(\sqrt{3}+\sqrt{2})\) (b) \(6(\sqrt{3}-\sqrt{2})\) (c) \(\frac{6}{(\sqrt{3}+\sqrt{2})}\) (b) \(6(\sqrt{3}-\sqrt{2})\) (d) none of these

Short answer

Question: Find the sum of the series: \(2\sqrt{3}, 2\sqrt{2}, \frac{4}{\sqrt{3}}, ...\) a) \(\frac{3\sqrt{3}(3^n-2)}{9}\) b) \(\frac{3\sqrt{2}(3^n-2)}{7}\) c) \(\frac{3\sqrt{6}(3^n-2)}{5}\) d) none of these Answer: d) none of these Explanation: The general expression for the sum \(S_n\) of the series up to n terms is \(S_n = \frac{2\sqrt{3}(1-(\frac{\sqrt{6}}{3})^n)}{1-\frac{\sqrt{6}}{3}}\), which doesn't match any of the given options.

Step-by-step solution

Identifying the common ratio

To find the common ratio between the terms in the series, divide the second term by the first term, and the third term by the second term. \(\frac{2\sqrt{2}}{2\sqrt{3}}=\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{6}}{3}\) \(\frac{4/\sqrt{3}}{2\sqrt{2}}=\frac{2}{3\sqrt{2}}=\frac{\sqrt{6}}{9}\) Since the ratios are equal, there is a common ratio, and it is \(\frac{\sqrt{6}}{3}\).

Applying the geometric series formula

The geometric series formula to find the sum of a series is: \(S_n = \frac{a(1-r^n)}{1-r}\) where \(S_n\) is the sum of the series, \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. In this case, \(a = 2\sqrt{3}\), and \(r = \frac{\sqrt{6}}{3}\). We are not given the number of terms, so we cannot use this formula directly to find the sum. However, we can find the general expression for the sum of the series up to n terms and then compare it with the given options.

Writing the general expression for the sum

Using the geometric series formula, we have: \(S_n = \frac{2\sqrt{3}(1-(\frac{\sqrt{6}}{3})^n)}{1-\frac{\sqrt{6}}{3}}\)

Comparing with the given options

Comparing the general expression for the sum \(S_n\) with the given options, we see that none of them match the expression we've derived. Therefore, the correct answer is (d) none of these.

Key concepts

Geometric Progression

When we talk about geometric progression (GP), we refer to a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the series given in the exercise, after identifying the common ratio as \(\frac{\sqrt{6}}{3}\), we can determine that the series is indeed a GP. Knowing that the series is a GP is key because it allows us to use specific formulas to find various properties of the series, such as the sum of the first 'n' terms.

To fully grasp the concept of GP, it's essential to become comfortable with the notion of the common ratio and how it repeats uniformly across the series. With practice, recognizing GPs and applying relevant formulas become second nature, aiding in solving related problems effectively.

Series and Sequences

Series and sequences are fundamental concepts in mathematics, particularly in the study of patterns and calculations over multiple terms. A sequence is an ordered list of numbers that follow a particular rule, while a series is the sum of the elements of a sequence. In the given problem, we're dealing with a series because we're interested in finding the sum of the terms in the sequence. The distinction between sequences and series is crucial for understanding how to approach problems and apply formulas correctly. For instance, the formula used in step 2 of the provided solution, \(S_n = \frac{a(1-r^n)}{1-r}\) is specifically for finding the sum of a geometric series up to 'n' terms, distinguishing it from merely listing the sequence of terms without summing them.

Square Roots in Series

Incorporating square roots into series can add complexity to problems, necessitating special attention to detail. Square roots often appear in geometric series, influencing the common ratio and thus the behavior of the series. As seen in this exercise, square roots factor into both the series terms and the common ratio. It's essential to manage them with care to ensure the accuracy of the common ratio and subsequent calculations.

It's useful to remember certain square root rules, such as \(\sqrt{a}\cdot\sqrt{b} = \sqrt{a\cdot b}\), when working with series involving square roots. Mastery of simplifying and manipulating expressions involving square roots is crucial for working through problems in this area effectively. Furthermore, being mindful that square roots can lead to both rational and irrational numbers is important as it can affect the final expression or sum of the series.