If h1, h2^' h3, …, h2 n be the 2 n harmonic means between a and b, then (h1+a)/(h1-a)+(h2 n+b)/

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If h1, h2^' h3, …, h2 n be the 2 n harmonic means between a...

If $\mathrm{h}_{1}, \mathrm{~h}_{2^{\prime}} \mathrm{h}_{3}, \ldots, \mathrm{h}_{2 \mathrm{n}}$ be the $2 \mathrm{n}$ harmonic means between a and $\mathrm{b}$, then $\frac{\mathrm{h}_{1}+\mathrm{a}}{\mathrm{h}_{1}-\mathrm{a}}+\frac{\mathrm{h}_{2 \mathrm{n}}+\mathrm{b}}{\mathrm{h}_{2 \mathrm{n}}-\mathrm{b}}$ (a) $\mathrm{n}$ (b) $2 \mathrm{n}$ (c) $4 n$ (d) $\underline{1}$

If $\mathrm{h}_{1}, \mathrm{~h}_{2^{\prime}} \mathrm{h}_{3}, \ldots, \mathrm{h}_{2 \mathrm{n}}$ be the $2 \mathrm{n}$ harmonic means between a and $\mathrm{b}$, then $\frac{\mathrm{h}_{1}+\mathrm{a}}{\mathrm{h}_{1}-\mathrm{a}}+\frac{\mathrm{h}_{2 \mathrm{n}}+\mathrm{b}}{\mathrm{h}_{2 \mathrm{n}}-\mathrm{b}}$
(a) $\mathrm{n}$
(b) $2 \mathrm{n}$
(c) $4 n$
(d) $\underline{1}$

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Key Concepts

Harmonic Mean

A harmonic mean of two numbers is defined as the reciprocal of the arithmetic mean of their reciprocals. It is used in situations where average rates are desired and plays a crucial role in forming harmonic sequences. Understanding the harmonic mean is fundamental when dealing with problems that involve harmonic means between two given numbers.

Harmonic Progression

A harmonic progression is a sequence of numbers in which the reciprocals form an arithmetic progression. This concept allows problems involving harmonic means to be transformed into problems involving arithmetic means, which are typically easier to manipulate algebraically. Recognizing a harmonic progression is key when working with sequences defined by such means.

Reciprocal Transformation

The method of reciprocal transformation involves converting a harmonic progression into an arithmetic progression by taking reciprocals. This technique simplifies the handling of harmonic sequences and is essential for deriving relationships between the terms. It is particularly useful in problems where the properties of arithmetic sequences can be exploited to establish invariants or summations.

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