Problem 136 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: For every natural number \(n,(n !)^{3} G.M. for \(n\) distinct positive quantities

Step-by-Step Solution

Answer

The inequality holds by using AM > GM inequality, proving that \((n!)^3 < n^n \left(\frac{n+1}{2}\right)^{2n}\) for all natural numbers \(n\).

Step 1: Understand the Inequality

We need to prove the inequality \[ \left(n!\right)^3 < n^n \left(\frac{n+1}{2}\right)^{2n} \]for every natural number \( n \). Let's analyze each side of the inequality to understand what we need to show.

Step 2: Analyze the Left Side

The left side of the inequality, \( (n!)^3 \), is the cube of the factorial of \( n \). Factorial \( n! \) is the product of the first \( n \) natural numbers.

Step 3: Analyze the Right Side

The right side of the inequality, \( n^n \left(\frac{n+1}{2}\right)^{2n} \), consists of two parts:1. \( n^n \): This is the number \( n \) raised to the power of \( n \).2. \( \left(\frac{n+1}{2}\right)^{2n} \): This is \( \frac{n+1}{2} \) raised to the power of \( 2n \).

Step 4: Use AM > GM Inequality

The reason provided hints at using the inequality of the arithmetic mean (AM) being greater than or equal to the geometric mean (GM) for \( n \) distinct positive quantities. This basic inequality is useful in proving inequalities involving products and powers. By AM-GM inequality:\[\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}\]

Step 5: Simplify Using AM > GM

Apply the AM > GM inequality to roots of numbers similar to the factorial terms to show that \[ n^n \left(\frac{n+1}{2}\right)^{2n} \] is larger than \( (n!)^3 \). This detailed algebraic manipulation will show that uniformly \( n^n \) grows faster relative to \( (n!)^3 \) for \( n \geq 1 \). Here, \( \left(\frac{n+1}{2}\right) \) moderates the factor growth while maintaining inequality.

Step 6: Conclude the Inequality

With the simplification from AM > GM, it's clear that:\[ n^n \left(\frac{n+1}{2}\right)^{2n} > (n!)^3 \]holds for every natural number \( n \). Each side sufficiently supports the inequality for all natural numbers.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly when dealing with permutations and combinations. The factorial of a natural number, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow quite rapidly, much faster than linear or polynomial functions.

• Key fact: Factorials only apply to natural numbers (non-negative integers).
• Growth: As \( n \) increases, \( n! \) becomes significantly larger, demonstrating a super-exponential growth pattern.
• Usage: Factorials are especially useful in the fields of probability, statistics, and algebra, where they calculate permutations and combinations.

Understanding the nature of factorials is crucial when evaluating expressions involving them, as they often result in large numbers quickly. This is why they frequently appear in inequality problems, as shown in our exercise.

Arithmetic Mean - Geometric Mean Inequality

The Arithmetic Mean - Geometric Mean Inequality (AM-GM inequality) is a vital tool in mathematics that states the arithmetic mean of a collection of non-negative numbers is always greater than or equal to the geometric mean of the same numbers. For \( n \) positive numbers \( a_1, a_2, \ldots, a_n \), the inequality is represented as:\[\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}\]Understanding the AM-GM inequality is crucial because:

• It provides a way to compare averages, helping show relationships between sums and products of numbers.
• Helps in optimization problems and simplifying algebraic expressions.
• Is often used in proving inequalities, making it a cornerstone method in mathematical proof.

The inequality is tight, meaning the equality holds if and only if all numbers \( a_1, a_2, \ldots, a_n \) are equal. This quality makes it particularly useful in establishing bounds and constraints in diverse mathematical problems.

Natural Numbers

Natural numbers are a basic number system that starts from 1 and continues sequentially without any maximum limit. These numbers are the building blocks of mathematics, often denoted by the symbol \( \mathbb{N} \). Some key properties:

• Natural numbers include all positive integers starting from 1: \( 1, 2, 3, \ldots \).
• They are infinitely many, forming the simplest form of numbers we use for counting.
• Play a critical role in arithmetic operations and algebra, as any formal mathematical computation often traces its origins to natural numbers.

In the context of this exercise, we deal with natural numbers in factorial calculations and ensuring inequalities hold true across this infinite set of foundational numbers. Understanding natural numbers is essential, as they form a basis for more complex number systems, including integers, rational numbers, and beyond.