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A progression is a sequence whose terms follow a certain pattern i.e., the terms are arranged under a definite rule.

Example: 1, 3, 5, 7, 9, …….. is a progression .

Types of Progression:

1. Arithmetic progression

2. Geometric progression

3. Arithmetico–geometric progression

4. Harmonic progression

5. Miscellaneous progressions

Arithmetic Progression (A.P.)

Definition:

Properties of an A.P. 

If  a, b, c, d,...... are in A.P. whose common difference is d, then for fixed non–zero number k ∈ R.

1. a ± k, b ± k, c ± k..... will be in A.P., whose common difference will be d.
2. ka, kab, kac..... will be in A.P. with common difference = kd.
3. a/k, b/k, c/k..... will be in A.P. with common difference = d/k.

Means if an Arithmetic progression is multiplied or divided by a constant it still remains an Arithmetic progression. But the common difference is changed, it is multiplied or divided by the constant respectively.

The sum of terms of an A.P. equidistant from the beginning and the end is constant and is equal to sum of first and last term. i.e. a₁ + a_n = a₂ + a_n–1 = a₃ + a_n–2....

If number of terms of any A.P. is odd, then sum of the terms is equal to product of middle term and number of terms.

Geometric Mean

If a, G, b are in G.P., then G is called G.M. between a and b. and G² = ab.

If a,G₁,G₂, G₃,......G_n, b then G₁,G₂, G₃,.....G_n are n geometric means between a and b.

Insertion of geometric means :

Single G.M. between a and b:

If a and b are two real numbers then single G.M. between a and b is given by G where G² = ab

n G.M.’s between a and b:  

a,G₁,G₂, G₃,....... G_n, b then G₁,G₂, G₃,....... G_n are n geometric means between a and b