Problem 19 - Algebra for JEE ADVANCED

Question

If \(a, b, c\), and \(d\) are four unequal positive numbers which are in A.P., then a. \(\frac{1}{a}+\frac{1}{d}>\frac{1}{b}+\frac{1}{c}\) b. \(\frac{1}{a}+\frac{1}{d}<\frac{1}{b}+\frac{1}{c}\) c. \(\frac{1}{b}+\frac{\mathrm{I}}{c}>\frac{4}{a+d}\) d. \(\frac{1}{a}+\frac{1}{d}=\frac{1}{b}+\frac{1}{c}\)

Step-by-Step Solution

Answer

Option a: \(\frac{1}{a} + \frac{1}{d} > \frac{1}{b} + \frac{1}{c}\).

Step 1: Understand the Given Condition

The numbers \(a, b, c, d\) are in Arithmetic Progression (A.P.). In an A.P., the difference between consecutive terms is constant. If \(a, b, c, d\) are in A.P., then \(b = a + n\), \(c = a + 2n\), and \(d = a + 3n\) where \(n\) is the common difference.

Step 2: Express in Reciprocal Terms

Calculate \(\frac{1}{a} + \frac{1}{d}\) and \(\frac{1}{b} + \frac{1}{c}\) using inverse expressions:\[\frac{1}{a} + \frac{1}{d} = \frac{1}{a} + \frac{1}{a + 3n} = \frac{a + 3n + a}{a(a + 3n)} = \frac{2a + 3n}{a^2 + 3an}\]\[\frac{1}{b} + \frac{1}{c} = \frac{1}{a + n} + \frac{1}{a + 2n} = \frac{a + 2n + a + n}{(a+n)(a+2n)} = \frac{2a + 3n}{a^2 + 3an + 2n^2}\]

Step 3: Compare the Two Expressions

Compare the fractions \(\frac{2a + 3n}{a^2 + 3an}\) and \(\frac{2a + 3n}{a^2 + 3an + 2n^2}\). The denominator \(a^2 + 3an\) is smaller than \(a^2 + 3an + 2n^2\), hence:\[\frac{2a + 3n}{a^2 + 3an} > \frac{2a + 3n}{a^2 + 3an + 2n^2}\]

Step 4: Conclude with the Correct Choice

Since \(\frac{1}{a} + \frac{1}{d} > \frac{1}{b} + \frac{1}{c}\), the correct answer is option a.

Key Concepts

Inequalities

Understanding inequalities is essential in comparing two values when there isn't a direct equivalence. An inequality shows either a "greater than" (>) or "less than" (<) relationship between two expressions. For example, if we say "x > y," it means that x is greater than y. In mathematics, inequalities are crucial for expressing ranges, limits, and optimizing solutions. They are often used in statistical predictions, economic modeling, and scientific computations.

When dealing with inequalities, it's important to remember that multiplying or dividing both sides by a negative number reverses the inequality sign. This rule is vital in solving inequality-based equations and must be carefully applied.

• For example: if -2x > 4, dividing both sides by -2 gives x < -2.

Inequalities in Arithmetic Progression (A.P.), like in this exercise, help us deduce relationships between elements. Here, the inequality between reciprocal expressions solidifies how positioning in an A.P. influences expression strength, providing deeper insights into number behaviors in sequences.

Reciprocal Expressions

Reciprocal expressions flip the roles of numerators and denominators in fractions. If a number is represented as a fraction with numerator and denominator, its reciprocal inverts these parts. For instance, the reciprocal of a number x is 1/x. This concept is key in various mathematical operations like division and equations set in fractional forms.

In our exercise, reciprocal expressions are used to compare terms in an Arithmetic Progression. This process involves examining the expressions:

• \( \frac{1}{a} + \frac{1}{d} \)
• \( \frac{1}{b} + \frac{1}{c} \)

Breaking these into common denominators, we use algebraic manipulations to derive comparisons. Solving reciprocal-oriented exercises often requires assessing how the reciprocal impacts magnitude and ordering, revealing potency or weakness in different placements within a sequence.

Mathematical Proofs

Mathematical proofs transform our understanding from assumptions to verified facts using logical reasoning. They form the backbone of mathematical rigor, affirming what is true through step-by-step deductions. In general terms, proofs may demonstrate properties of numbers, validate equations, or verify complex theories.

In our problem, proving the inequality involves several stages:

• Setting expressions from an A.P. context: identify terms and relate them using the common difference.
• Ensuring correct reciprocal conversion of terms to uncover insights about their equation relationships.
• Applying logical analysis to determine the inequality's validity based on comparative denominators.

Each proof's conclusion is pivotal; it shows that logical exposition confirms our initial hypothesis or directs us to review assumptions when the results differ. This disciplined approach builds necessary mathematical skills, enhancing your ability to tackle extensive and intricate mathematical challenges.