In this post we will explore the concept of progression for solving the problems based on Arithmetic, Geometric and Harmonic Progressions. Starting with definitions we will move into important formulaes and tricks to solve the problem easily and in short time frame

To get the idea on how the AP, GP and HP are linked please go here.

To read on the Harmonic Conjugate please read here.

**Arithmetic progressions:** Sequence of numbers where the difference between the consecutive numbers is constant

so we can say a_{n }– a_{n-1} = d where n is a postive integer and d is constant, it is also called as common difference

For the 3 consecutive numbers the middle number is the mean of other two i.e

Also for all n, m, l, >= 1 $ we will always have Arithmetic Progression is given as

Sum of the n of terms in Arithmetic Progressions is given as S = (n/2) x (first term + last term) if d = common difference

The arithmetic mean is represented as

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**Geometric Progressions**: Sequence of numbers where the ratio between the consecutive numbers is constant

Thus a_{n}, a_{n-1} and a_{n-2} are in geometric progression GP in short the a_{n} = r x a_{n-1 }r = common ratio

The sum of a GP with n terms and r common ratio is given as

For any three consecutive terms we can say a_{l}, a_{m} and a_{n}

Please note the sequence on m and l in the power

Geometric mean is represented as

** **

**Harmonic Progressions: **A progressions formed by reciprocal of Arithmetic Progressions represendted as

The harmonic mean is given as

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