If √(b)+√(c), √(c)+√(a), √(a)+√(b) are in HP, then a, b, c are in (a) AP (b) HP (c) GP (d) AGP

Name: Problem 57, Chapter 5: IIT-JEE Super Course in Mathematics
Uploaded: 2021-07-03T18:09:56-08:00
Duration: 1 min 18 s
Channel: Narayan Hari
Description: If √(b)+√(c), √(c)+√(a), √(a)+√(b) are in HP, then a, b, c are in (a) AP (b) HP (c) GP (d) AGP

step 1 In this problem, I can write the value of 1 by under root A plus root B plus 1 by under root B p

If √(b)+√(c), √(c)+√(a), √(a)+√(b) are in HP, then a, b, c...

If $\sqrt{\mathrm{b}}+\sqrt{c}, \sqrt{\mathrm{c}}+\sqrt{\mathrm{a}}, \sqrt{\mathrm{a}}+\sqrt{\mathrm{b}}$ are in $\mathrm{HP}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in
(a) $\mathrm{AP}$
(b) HP
(c) $\mathrm{GP}$
(d) AGP

Key Concepts

Harmonic Progression (HP)

A sequence of numbers is said to be in harmonic progression if their reciprocals form an arithmetic progression. This key concept is often used to solve problems by converting a harmonic progression into an arithmetic progression by taking reciprocals, which simplifies the algebraic manipulation and reveals underlying relationships among the terms.

Arithmetic Progression (AP)

An arithmetic progression is a sequence in which the difference between consecutive terms is constant. In many problems, establishing that certain expressions satisfy the AP condition (for example, through showing that the sum of the first and third terms equals twice the second term) is a critical step in deducing relationships among the variables involved.

Transformation of Sequences

Transforming sequences through operations such as taking reciprocals or squaring the terms is a useful method in sequence problems. For instance, converting expressions involving square roots into squared terms can simplify relationships, translating a condition on square root sums into a condition on the original variables, which often leads to identifying a progression type such as AP or GP.

Reciprocal Relationships

Understanding the relationship between a sequence and the sequence of its reciprocals is essential, especially when dealing with harmonic progressions. Recognizing that the reciprocals of the terms in a harmonic progression form an arithmetic progression enables one to derive conditions such as the sum of the first and third terms equaling twice the second term, which can then be manipulated to reveal further properties of the original sequence.

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