**Harmonic Progression:** 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

Hence 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 \(\frac{1}{8}\).

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