Chapter 27: Harmonical Progression

A series of quantities is said to be in Harmonical Progression (H.P.) if their reciprocals form an Arithmetical Progression. For example, if 1/2, 1/4, 1/6 are in A.P., then 2, 4, 6 are in H.P.

27.1 Properties of Harmonical Progression

Unlike A.P. and G.P., there is no simple formula for the sum of a Harmonical series. To solve problems involving H.P., we generally follow these steps:

  1. Take the reciprocals of the given terms.
  2. Solve the resulting problem using Arithmetical Progression formulas.
  3. Invert the result to return to the Harmonical scale.
Example: Find the 10th term of the H.P. 2/3, 1/2, 2/5, ...

Step 1: Take reciprocals to find the A.P.
Reciprocals: 3/2, 2, 5/2... (Common difference d = 1/2).

Step 2: Find the 10th term of this A.P.
l = a + (n-1)d = 3/2 + (9)(1/2) = 12/2 = 6.

Step 3: Invert the result.
Result: The 10th term of the H.P. is 1/6.

27.2 Harmonical Mean

The Harmonical Mean (H) between two numbers a and b is found using the formula:

H = 2ab / (a + b)

27.3 Relation between Means

For any two positive numbers, there is a distinct relationship between their Arithmetical (A), Geometrical (G), and Harmonical (H) means:

A × H = G²

This implies that G is the Geometrical Mean between A and H.

Physics Connection: The Harmonical Mean is used to calculate "average speed" over a distance when the speeds for each half are different. It is also the formula used to find the equivalent resistance of resistors in parallel or the focal length of lenses.
Doctor's Pro-Tip: Don't try to memorize new formulas for H.P. just to find the nth term. Always "flip it" to A.P., do the work you already know how to do, and "flip it" back!