Harmonic progression

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A harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an Arithmetic Progression. In other words, it is a sequence of the form

 

where −1/d is not a natural number and k is a natural number.

 

Harmonic Mean

The harmonic mean of two numbers is the reciprocal of arithmetic means of the reciprocals of the two numbers. Typically, it is appropriate for situations when the average of rates is desired. Harmonic mean of 2 numbers, a and b can be found using the formula

 

Harmonic Mean of more than two numbers can be found using the following formula

 

 

It can also be denoted in terms of the geometric mean and arithmetic mean as

 

Applications of Harmonic Mean

  1. Average Speed - If a vehicle travels a certain distance at a speed x (40 kilometres per hour) and then the same distance again at a speed y (60 kilometres per hour), then its average speed is the harmonic mean of x and y (48 kilometres per hour). However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at a speed y, then its average speed is the arithmetic mean of x and y, which in the above example is 50 kilometres per hour. The same principle applies to a series of sub-trips at different speeds. If each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds.
  2. In any triangle, the radius of the incircle is one-third the harmonic mean of the altitudes.
  3. For any point P on the minor arc BC of the circumcircle of an equilateral triangle ABC, with distances q and t from B and C respectively, and with the intersection of PA and BC being at a distance y from point P, we have that y is half the harmonic mean of q and t.
  4. In a right triangle with legs a and b and altitude h from the hypotenuse to the right angle, h2 is half the harmonic mean of a2 and b2.
  5. Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be the intersection of the diagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.)

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