Problem solving with rules of progressions

This post bring in some important theorems and rules that help to solve the problems of progressions.

the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM)  for non-negative numbers will follows:

AM >= GM >= HM

Also for the given set of 2 positive numbers

AM x HM =  GM²

For reference the Arithmetic means, Geometric Mean and Harmonic mean is give as where x and y are positive numbers

IMPORTANT : if a sequence is picked from a progression then it follows the same properties of sequence.  For example, if there is a sequence is 2, 4, 6, 8 ……. 200  this sequence has first term a =2 and common difference = 2, if we pick up a sequence like a₁₀, a₂₀, a₃₀ then sequence is also an AP with common difference is 20

This is true for all type of Sequence AP, GP and HP

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