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# The harmonic mean between two numbers a and b isA.$\dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}$B.$\dfrac{a b}{\dfrac{1}{a}+\dfrac{1}{b}}$C.$\dfrac{2 n}{\dfrac{1}{a}+\dfrac{1}{b}}$D.$\dfrac{1}{\dfrac{1}{a}-\dfrac{1}{b}}$

Last updated date: 15th Sep 2024
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Hint: The harmonic mean of two numbers is in fact the reciprocal of arithmetic mean of the reciprocal of the numbers. This simply means that if $\mathrm{H}$ is the harmonic mean between two numbers say a and $\mathrm{b}$ then $1 / \mathrm{a}, 1 / \mathrm{H}$ and $1 / \mathrm{b}$ are in A.P. Harmonic Mean is the reciprocal of the arithmetic mean of the reciprocals. In other words, it is the number of observations, divided by the sum of reciprocals of the observations. This also is one of several kinds of average and it is appropriate for situations when the average of rates is desired. The mean, also referred to by statisticians as the average, is the most common statistic used to measure the center of a numerical data set. The mean is the sum of all the values in the data set divided by the number of values in the data set.

Harmonic mean $=\dfrac{\mathrm{n}}{\sum_{\mathrm{i}=1}^{\mathrm{n}} \dfrac{1}{\mathrm{x}_{1}}}=\dfrac{2}{\left(\dfrac{1}{\mathrm{a}}+\dfrac{1}{\mathrm{b}}\right)}$
Note: Harmonic mean has the least value among all the three means. The relationship between arithmetic mean, geometric mean and harmonic mean is: "The product of arithmetic mean and harmonic mean of any two numbers a and $b$ in such a way that $a>b>0$ is equal to the square of their geometric mean." $A M \times H M=G M^{2}$. For two numbers $x$ and y, let x, a, y be a sequence of three numbers. If x, a, y is an arithmetic progression then 'a' is called arithmetic mean. If $\mathrm{x}, \mathrm{a}, \mathrm{y}$ is a geometric progression then 'a' is called geometric mean. If $\mathrm{x}, \mathrm{a}, \mathrm{y}$ form a harmonic progression then 'a' is called harmonic mean.