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 = GM ^{2}**

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 a_{10}, a_{20}, a_{30} 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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