Harmonic mean

What is Harmonic Mean?
Formula for harmonic mean (explained below)
When three numbers are in harmonic progression, the middle number is called the Harmonic mean of the other two numbers.

For example, if a, b and c are in harmonic progression, the middle number c is called the harmonic mean of a and b.

What is the formula for calculating Harmonic mean?
The formula for calculating harmonic mean cannot be directly obtained by taking a harmonic progression. It is obtained by taking the reciprocals of a harmonic progression, which gives us an arithmetic progression. Then it becomes easier to calculate the harmonic mean as described below:

Let three numbers a, b and c be in harmonic progression. Hence b is the harmonic mean between a and b. Now, the reciprocals of these numbers,
1/a, 1/b, 1/c
are in arithmetic progression (because a harmonic progression is defined as the sequence of numbers obtained by taking the reciprocals of each number of an arithmetic progression)

Now, since 1/b is the arithmetic mean between 1/a and 1/c, thus
1/b = (1/a + 1/c)/2
(the above result is obtained by the formula for arithmetic mean)

On solving the above equation, we get
b = 2ac/(a + c)
Therefore the formula for harmonic mean between two numbers a and c is
b = 2ac/(a + c), where b is the harmonic mean between a and c
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