Problem 3 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the sum of the first 20 terms of the series \(\log _{\left(7^{1 / 2}\right)} x+\log _{\left(7^{1 / 3}\right)} x+\log _{\left(7^{1 / 4}\right)} x+\ldots\) is 460 , then \(x\) is equal to : [Sep. 05, 2020 (II)] (a) \(7^{2}\) (b) \(7^{1 / 2}\) (c) \(e^{2}\) (d) \(7^{46 / 21}\)

Step-by-Step Solution

Answer

x = 7^{46/21} (Option d)

Step 1: Understanding the Series

The given series is a sum of logarithms with different bases:\[ S = \log_{7^{1/2}} x + \log_{7^{1/3}} x + \log_{7^{1/4}} x + \ldots \] Each term in this series can be written as a logarithm with a base of form \( 7^{1/n} \). Thus, the \(n\)-th term can be written as \( \log_{7^{1/n}} x \).

Step 2: Express Each Logarithm in a Common Base

We know the change of base formula: \[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]Applying the change of base formula, each term becomes:\[ \log_{7^{1/n}} x = \frac{\log x}{\log 7^{1/n}} = \frac{n \log x}{\log 7} \]By simplifying, we find the expression for the \(n\)-th term of the series.

Step 3: Sum the Series for the First 20 Terms

Now, let's sum up the series for the first 20 terms:\[ S = \frac{1 \cdot \log x}{\log 7} + \frac{2 \cdot \log x}{\log 7} + \ldots + \frac{20 \cdot \log x}{\log 7} \]This simplifies to:\[ S = \frac{\log x}{\log 7} (1 + 2 + 3 + \ldots + 20) \]The sum of the first 20 natural numbers is \( \frac{20 \cdot 21}{2} = 210 \). Hence:\[ S = \frac{210 \cdot \log x}{\log 7} \]

Step 4: Equate the Series to 460

We are given that the sum \( S = 460 \). So we can set up the equation:\[ \frac{210 \cdot \log x}{\log 7} = 460 \]Solving this equation for \( \log x \), we have:\[ 210 \cdot \log x = 460 \cdot \log 7 \]\[ \log x = \frac{460}{210} \cdot \log 7 \]

Step 5: Solve for x

To solve for \( x \), write:\[ x = 7^{\frac{460}{210}} \]We simplify \( \frac{460}{210} \) to \( \frac{46}{21} \). Thus:\[ x = 7^{\frac{46}{21}} \]

Step 6: Conclusion

Comparing with the answer choices, we find that the solution matches option (d). Hence, \( x = 7^{46/21} \).

Key Concepts

Sum of Series

In mathematical series, calculating the sum is often a crucial task. In this problem, we are dealing with a series comprised of logarithmic expressions. Each term in the series is a logarithm, with the base being a fractional power of 7. The goal is to find the sum of the first 20 terms. Here's how you usually find the sum of a series:

• Identify each term in the series.
• Use any relevant formulas to express the terms in a simpler form.
• Add up all the terms to get the overall sum.

In the context of the given problem, the series sums up all the logarithmic terms from the first to the twentieth. To simplify finding the sum of these terms, we use the formula for the sum of the first n natural numbers, which is \( \frac{n(n+1)}{2} \). This formula is applied to the series that adds efficiently once the terms are placed in a common base using the change of base formula.

Change of Base Formula

The change of base formula is a fundamental tool in logarithmic calculations, allowing us to convert logarithms from one base to another. This is particularly useful when dealing with series or expressions that have different bases, as seen in the original exercise.The formula is expressed as:\[\log_{b} a = \frac{\log_{c} a}{\log_{c} b}\]This means you can express any logarithm in terms of a new base by dividing the logarithm of the number and the logarithm of the original base, both calculated with the new base.In our problem, each term of the series involves a logarithm with a base that is a root of 7, such as \( 7^{1/2}, 7^{1/3}, \) and so on. We convert all these bases to a common base, say base 7, using the change of base formula. This conversion simplifies the process of adding the series because it allows all terms to be expressed as multiples of \( \log x \) divided by \( \log 7 \). This commonality is what makes the subsequent calculation steps feasible.

Mathematical Problem Solving

Mathematical problem solving involves a structured process to find solutions logically. In this exercise, solving the problem is about breaking it down into manageable steps and using mathematical identities and properties, such as those of logarithms.Here is a simple approach:

• Begin by understanding the nature of the problem and what it is asking for. In this case, it involves finding a specific value of \( x \) given a sum of logarithmic terms.
• Deconstruct complex expressions using relevant formulas, such as the change of base formula for logarithms, which allows for simplification.
• Organize your calculations and strategically use mathematical equations to solve for the unknown variable, ensuring all steps follow logically.
• Finally, compare your findings to existing options if provided, and conclude whether your solution aligns with one of them; this verifies the accuracy of your work.

Approaching problems with these strategies not only aids in solving the current problem but also builds the skills necessary to tackle similar problems in the future.