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Harmonic Progression

Unequal numbers a, b, c,…are said to be in a harmonic progression HP if reciprocals of these terms i.e., Description: 2597.pngare 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 HP = 1/(nth term of the corresponding AP)
If a, b and c are in HP, then Description: 2606.png Description: 2615.png are in
  1. AP
  2. GP
  3. HP
  4. Cannot be determined uniquely
abc are in HP, so,Description: 2624.pngwill be in AP.
Or, Description: 2633.pngwill be in AP.
Or, Description: 2642.png will be in AP.
Hence Description: 2651.png are in AP.
So, Description: 2660.pngwill be in HP.
Alternatively, these kinds of problems can also be done by assuming values.
Let us take 1, ½ , 1/3 (which are in HP)
Description: 2669.pngDescription: 2678.pngDescription: 2688.png
Now, when we check these values for AP, GP and HP, we find thatDescription: 2697.pngis the AM ofDescription: 2706.pngandDescription: 2715.png
So, obviously the given terms are in HP.

Properties of HP

If a, b, c and d are in HP, then
  1. a + d > b + c
  2. 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.

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