# Harmonic Progression: Introduction

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 $$\frac{1}{8}$$.