The terms of a series whose reciprocal form an arithmetic progression is called harmonic progresion.
Example: Find out if the given series is a harmonical series? $$ 1, \frac{1}{3}, \frac{1}{5}, \frac{1}{7},.... $$ Here the reciprocals of the series 1, 3, 5, 7... are in arithmetic progression, hence the given series is a harmonic series.
Note: We can use all the arithmetic series formulas to find the \(n^{th}\) term of a harmonical series and harmonic means after the reciprocal of a harmonic series. Check the examples given below and Click here to get the arithmetic progression notes.
Example(1): Find the fifth term of the given harmonic series? $$ \frac{1}{2}, \frac{1}{4}, \frac{1}{6},.... $$
Solution: The reciprocals of the given series are in arithmetic progression. $$ 2, 4, 6,.... $$ Here a = 2 and d = 4 - 2 = 2
The fifth term of the series. $$ T_n = a + (n - 1)d $$ $$ T_5 = 2 + (5 - 1) \ 2 $$ $$ T_5 = 10 $$ Here 10 is the fifth term of the arithmetic series \(2, 4, 6,....\)
Hence the fifth term of the harmonic series will be \(\frac{1}{10}\).
Example(2): Find the harmonic mean of \(\frac{1}{4}\) and \(\frac{1}{8}\)?
Solution: Reciprocal of the harmonic progression terms are 4 and 8.
Here a = 4 and b = 8
According to the arithmetic mean formula $$ M = \frac{1}{2} \ (a + b) $$ $$ M = \frac{1}{2} \ (4 + 8) $$ $$ M = 6 $$ Here 6 is the arithmetic mean of 4 and 8.
Hence \(\frac{1}{6}\) is the harmonic mean of \(\frac{1}{4}\) and \(\frac{1}{8}\).