Question

The harmonic mean between the two numbers a and b is
(a) \[\dfrac{a+b}{2ab}\]
(b) \[\dfrac{a-b}{2ab}\]
(c) \[\dfrac{2ab}{a+b}\]
(d) \[\dfrac{2ab}{a-b}\]

Answer 1

Hint: We will use the formula of harmonic mean to solve this question. Harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals. The formula to calculate the harmonic mean is given by \[=\dfrac{n}{\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}..............}\].

Complete step-by-step answer:
Before proceeding with the question, we should understand the concept of harmonic progression and harmonic mean.
A Harmonic Progression (HP) is defined as a sequence of real numbers which is determined by taking the reciprocals of the arithmetic progression that does not contain 0. In the harmonic progression, any term in the sequence is considered as the harmonic means of its two neighbors. For example, the sequence a, b, c, d, …is considered as an arithmetic progression, the harmonic progression can be calculated as \[\dfrac{1}{a}\], \[\dfrac{1}{b}\], \[\dfrac{1}{c}\], \[\dfrac{1}{d}\], …
Harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals. The formula to calculate the harmonic mean is given by:
Harmonic mean \[=\dfrac{n}{\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}..............}......(1)\] where a, b, c, d are the values and n is the number of values present.
So here in the question we have to find the harmonic mean between two numbers a and b. So in this case n is equal to 2. So using formula (1) we get,
Harmonic mean between two numbers a and b \[=\dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}......(2)\]
Now solving the denominator part in the equation (2) by taking the LCM we get,
\[\Rightarrow \dfrac{2}{\dfrac{a+b}{ab}}......(3)\]
Now rearranging the denominator in equation (3) we get,
\[\Rightarrow \dfrac{2ab}{a+b}\]
Hence the harmonic mean between two numbers is \[\dfrac{2ab}{a+b}\]. So the correct answer is option (c).

Note: Remembering the general formula of harmonic mean for n values is the key here because we may think option (d) as the answer. Also in a hurry we can make a mistake in solving equation (2) and equation (3) so we need to be careful while doing this step.