Problem 72 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(A_{n}=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^{2}+\left(\frac{3}{4}\right)^{3}-\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^{n}\) and \(B_{n}=1-A_{n}\). Then, the least odd natural number \(p\), so that \(B_{n}>A_{n}\), for all \(n \geq p\) is [Online April 15, 2018] (a) 5 (b) 7 (c) 11 (d) 9

Step-by-Step Solution

Answer

The least odd natural number is 7.

Step 1: Understand the Sequence Structure

We first note that the sequence \(A_n\) is given as:\[A_n = \left(\frac{3}{4}\right) - \left(\frac{3}{4}\right)^{2} + \left(\frac{3}{4}\right)^{3} - \ldots + (-1)^{n-1}\left(\frac{3}{4}\right)^{n}\]This is an alternating series with common ratio \(\frac{3}{4}\) and alternating sign to each power.

Step 2: Transform to Summation

The expression for \(A_n\) can be transformed into a geometric series formula:\[A_n = \sum_{k=1}^{n} (-1)^{k-1} \left(\frac{3}{4}\right)^{k}\]This series alternates between positive and negative terms.

Step 3: Geometric Series Formula

To find the sum of the geometric series, we use the formula for the sum of the first \(n\) terms of an alternating series:\[A_n = \frac{\frac{3}{4} \left( 1 - (-\frac{3}{4})^n \right)}{1 - (-\frac{3}{4})}\]

Step 4: Simplify the Expression for A_n

The simplification of the expression for \(A_n\) becomes:\[A_n = \frac{\frac{3}{4} \left( 1 - (-1)^n \left(\frac{3}{4}\right)^n \right)}{\frac{7}{4}}\]Simplifying further gives:\[A_n = \frac{3}{7} \left(1 - (-1)^n \left(\frac{3}{4}\right)^n \right)\]

Step 5: Evaluate B_n

We have \(B_n = 1 - A_n\), so:\[B_n = 1 - \frac{3}{7} \left(1 - (-1)^n \left(\frac{3}{4}\right)^n \right)\]

Step 6: Simplify B_n Expression

Further simplifying \(B_n\), we get:\[B_n = \frac{7}{7} - \frac{3}{7} + \frac{3}{7} (-1)^n \left(\frac{3}{4}\right)^n\]This results in:\[B_n = \frac{4}{7} + \frac{3}{7} (-1)^n \left(\frac{3}{4}\right)^n\]

Step 7: Analyze Inequality B_n > A_n

To find \(p\), we solve the inequality:\[\frac{4}{7} + \frac{3}{7} (-1)^n \left(\frac{3}{4}\right)^n > \frac{3}{7} \left(1 - (-1)^n \left(\frac{3}{4}\right)^n \right)\]Simplifying, this reduces to:\[4 > 0 + (-1)^n \left(\frac{3}{4}\right)^n\]For odd \(n\), \((-1)^n = -1\) and for even \(n\), \((-1)^n = 1\). Consider odd \(n\), the inequality becomes \(\left(\frac{3}{4}\right)^n < 1\).

Step 8: Find Minimum Odd n

Given \((\frac{3}{4})^n < 1\) simplifies to finding the smallest \(n\) where\[((\frac{3}{4})^n) < 1\] for \(n\) odd.By evaluating powers: \((\frac{3}{4})^5 \approx 0.3164\) and \((\frac{3}{4})^7 \approx 0.178\) shows \((\frac{3}{4})^7 < 1\) whereas for \((\frac{3}{4})^5 < 1\), indicating they switch. Thus, \(n = 7\) is confirmed correct.

Key Concepts

Geometric Series

A geometric series is 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. In the sequence given in the exercise, the terms \[A_{n} = \left(\frac{3}{4}\right) - \left(\frac{3}{4}\right)^{2} + \left(\frac{3}{4}\right)^{3} - \ldots + (-1)^{n-1}\left(\frac{3}{4}\right)^{n}\]are part of a geometric sequence with a common ratio of \(\frac{3}{4}\). This ratio is significant because it tells us how the sequence is progressively multiplied, creating a pattern:

• Each term is formed by multiplying the previous term by \(\frac{3}{4}\).
• The sign alternates with each term due to the \((-1)^{n-1}\) factor, making it an **alternating** geometric series.

Geometric series are commonly explored through problems involving their sum. The sum of a geometric series can be computed with specific formulas that are adjusted for infinities or finite terms, providing major insights into how a sequence behaves as it extends.

Series Inequality

When we have two sequences or series, a typical exercise might involve comparing them, often with an inequality. In this particular exercise, the focus is on the inequality:\[B_{n} > A_{n}\]Here, determining when this inequality holds involves evaluating the sequences at different values of \(n\) and comprehending the behavior of the terms:

• The inequality directly compares the summed terms of the sequences.
• To resolve it, the sequences are expressed as functions of \(n\), enabling comparison.
• Analysis shows that changing values of \(n\), especially when odd, affects the alternating factors significantly.

In inequalities involving series, the largest contributing factor is often how the terms grow or shrink, particularly influenced by the common ratio and sign in geometric series. Ensuring comprehension of these relationships is critical to resolving such inequalities.

Sum of Series

Understanding the sum of a series like the one in this exercise is pivotal in solving problems effectively. For the series\[A_n = \sum_{k=1}^{n} (-1)^{k-1} \left(\frac{3}{4}\right)^{k}\]we use specific sum formulas for geometric series to find its overall value.

• The alternating sign is handled by adjusting typical sum equations, modifying the geometric series sum formula.
• Typically, the sum for an infinite geometric series where the absolute value of the ratio is less than 1 is given as \(S = \frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the ratio.
• Derivations adapt these formulas for finite terms which is crucial for finding precise sums like that of finite, alternating series.

The application of these formulas allows us to simplify complex series into manageable expressions, guiding towards solutions like confirming that \(n = 7\) is the minimum odd number such that \(B_n > A_n\).