# Harmonic Progression

Unequal numbers*a, b, c*,â€¦are said to be in a harmonic progression HP if reciprocals of these terms i.e., are in an AP. It is noteworthy, that no term of a HP can be equal to zero.

nth term of a harmonic progression

nth term of a harmonic progression

*n*th term of HP = 1/(

*n*th term of the corresponding AP)

Example-12

If

*a, b*and*c*are in HP, then are in- AP
- GP
- HP
- Cannot be determined uniquely

Solution

*a*,

*b*,

*c*are in HP, so,will be in AP.

Or, will be in AP.

Or, will be in AP.

Hence are in AP.

So, will be in HP.

Alternatively, these kinds of problems can also be done by assuming values.

Let us take 1, Â½ , 1/3 (which are in HP)

, ,

Now, when we check these values for AP, GP and HP, we find thatis the AM ofand

So, obviously the given terms are in HP.

# Properties of HP

If a, b, c and d are in HP, then*a*+*d*>*b*+*c**ad*>*bc*

*Sum of n terms of a harmonic progression*

There is no standard formula for finding the sum of

*n*terms of a HP.