Question

Two AM's \(\mathrm{A}_{1}\) and \(\mathrm{A}_{2}\), two GM's \(\mathrm{G}_{1}\) and \(\mathrm{G}_{2}\) and two \(\mathrm{HM}\) 's \(\mathrm{H}_{1}\) and \(\mathrm{H}_{2}\) are inserted between two positive numbers. Then, \(\mathrm{H}_{1}^{-1}+\mathrm{H}_{2}^{-1}=\) (a) \(\mathrm{A}_{1}^{-1}+\mathrm{A}_{2}^{-1}\) (b) \(\mathrm{G}_{1}^{-1}+\mathrm{G}_{2}^{-1}\) (c) \(\mathrm{A}_{1} \mathrm{H}_{1}+\mathrm{A}_{2} \mathrm{H}_{2}\) (d) \(\frac{A_{1}+A_{2}}{G_{2} G_{2}}\)

Short answer

Answer: (d) \(\frac{A_1+A_2}{G_1G_2}\)

Step-by-step solution

Understand the properties of AM, GM, and HM

Arithmetic Mean (AM) is the sum of numbers divided by the total count of numbers. Geometric Mean (GM) is the nth root of the product of n numbers. Harmonic Mean (HM) is the reciprocal of the arithmetic mean of the reciprocals of the given numbers. Given two numbers a and b: - AM = \((a + b)/2\) - GM = \(\sqrt{ab}\) - HM = \(2ab/(a + b)\)

Deriving the relationship between AM's, GM's, and HM's

We are given two AM's (\(A_1\) and \(A_2\)), two GM's (\(G_1\) and \(G_2\)), and two HM's (\(H_1\) and \(H_2\)) between two positive numbers. Let's denote these positive numbers as \(a\) and \(b\). We can write: - \(A_1 = (a + x)/2\), \(A_2 = (b + x)/2\) - \(G_1 = \sqrt{ax}\), \(G_2 = \sqrt{bx}\) - \(H_1 = 2ax/(a + x)\), \(H_2 = 2bx/(b + x)\)

Finding the value of \(H_1^{-1} + H_2^{-1}\)

Now we need to find the value of \(H_1^{-1} + H_2^{-1}\). Using the definitions of \(H_1\) and \(H_2\) from step 2, we get: \(H_1^{-1} + H_2^{-1} = \frac{a + x}{2ax} + \frac{b + x}{2bx} = \frac{(a + x)b + (b + x)a}{2abx}\) Now, simplifying: \(H_1^{-1} + H_2^{-1} = \frac{ab + ax + ba + bx}{2abx} = \frac{2ab + (a + b)x}{2abx}\) Now, we can observe that \(a + x = 2A_1\) and \(b + x = 2A_2\) from the definitions of \(A_1\) and \(A_2\), allowing us to rewrite the expression as: \(H_1^{-1} + H_2^{-1} = \frac{2ab + (2A_1 + 2A_2)x}{2abx} = \frac{A_1 + A_2}{G_1G_2}\) So, the answer is (d) \(\frac{A_1+A_2}{G_1G_2}\).

Key concepts

AM GM HM Inequalities

In mathematics, specifically in the field of algebra, the mean of a set of numbers is a way to express the central value of the set. There are different types of means, each with their unique properties. Among the most commonly encountered means are the Arithmetic Mean (AM), the Geometric Mean (GM), and the Harmonic Mean (HM). These means are not only core concepts in mathematics but also critical in understanding their inequalities, which form the basis of the AM-GM-HM Inequality.

The AM-GM-HM Inequality is a fundamental relationship in algebra that provides a comparison between the arithmetic, geometric, and harmonic means of any non-negative real numbers. It states that for any set of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean. Mathematically, for any non-negative numbers \(a_1, a_2, ..., a_n\), the inequality is expressed as:

\[ \frac{a_1 + a_2 + ... + a_n}{n} \geq \sqrt[n]{a_1 \times a_2 \times ... \times a_n} \geq \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ... + \frac{1}{a_n}} \]

This relationship becomes an essential tool in problem-solving, especially when estimating or comparing the sizes of certain quantities and is particularly valued in competitive exams like the IIT-JEE, where maximizing efficiency is of the essence.

In our original exercise, understanding the AM-GM-HM relationships is crucial in deriving the relationship between the inserted means and finding the correct answer.

Sequences and Series

Sequences and series are foundational concepts in mathematics, particularly in calculus and analysis. A sequence is an ordered list of numbers where each number is called a term. A series, on the other hand, is the sum of the terms of a sequence.

In the realm of sequences, there are arithmetic sequences where the difference between consecutive terms is constant, and geometric sequences where the ratio of consecutive terms is constant. To understand the application of sequences in our problem, it's important to note that inserting AMs between two numbers creates an arithmetic sequence, whereas inserting GMs creates a geometric sequence.

Furthermore, in the context of the original problem provided, the AMs, GMs, and HMs inserted between the two positive numbers can be thought of as terms within these arithmetic and geometric sequences. Bridging sequences and series with the concept of means allows us to solve for variables and establish relationships between different terms in more complex mathematical contexts, such as in the provided exercise, where we're examining inverses and their summations in relation to the inserted means.

IIT-JEE Mathematics

IIT-JEE Mathematics is renowned for its challenging and comprehensive syllabus, which tests students' problem-solving skills and understanding of various mathematical concepts, including algebra, calculus, trigonometry, and more. The IIT-JEE, formally known as the Joint Entrance Examination, is an engineering entrance assessment conducted in India, considered to be one of the toughest exams in the world.

Problems related to means, such as those involving AM, GM, and HM inequalities, play a significant role in the algebra section of the IIT-JEE Mathematics syllabus. Excelling in these areas requires not only familiarity with the formulas and properties but also the ability to apply them creatively in novel situations.

The exercise provided is typical of the problems that might be encountered by students preparing for the IIT-JEE. It requires a clear understanding of both arithmetic and geometric sequences and their means, as well as the ability to manipulate and combine these concepts to arrive at the solution. Students aiming to crack the IIT-JEE must develop the skills to approach such problems methodically, something that gets honed with significant practice and targeted learning strategies.