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Harmonic Progression (H.P.)

Definition:

A progression is called a harmonic progression (H.P.) if the reciprocals of its terms are in A.P.

Standard form: 1/a, 1/(a + d), 1/(a + 2d)......1, 1/2, 1/3,....... are in harmonic progression and 1, 2, 3..... are in arithmetic progression.

General term of an H.P.

Harmonic Mean:

If three or more numbers are in H.P., then the numbers lying between the first and last are called harmonic means (H.M.’s) between them. For example 1, 1/3, 1/5, 1/7, 1/9 are in H.P. So 1/3, 1/5 and 1/7 are three H.M.’s between 1 and 1/9.

Also, if a, H, b are in H.P., then H is called harmonic mean between a and b.

Arithmetico-geometric Progression (A.G.P.)

Definition:

The combination of arithmetic and geometric progression is called arithmetico–geometric progression.

n^th term of A.G.P.

If a₁, a₂, a₃, a₄,.....an is an A.P.

b₁, b₂, b₃....b_n is a G.P.,

then the sequence a₁b₁, a₂b₂, a₃b₃,.......a_nb_n is said to be an arithmetico–geometric sequence.

Thus, the general form of an arithmetico geometric sequence is;

a, (a + d)r, (a + 2d)r², (a + 3d)r³.......a + (n – 1)dr^(n–1)

Sum of A.G.P.

Point:

A point is that which has no part. A point is represented by a fine dot made by a sharp pencil on a sheet of paper.

Plane:

The surface of a smooth wall or the surface of a sheet of paper or the surface of a smooth black board is close examples of a plane.

Line:

A line is breadth less length e.g., the edge of a ruler, the edge of the top of a table, the meeting place of two walls of a room is also examples of a geometrical straight line.

Collinear Points:

Three or more points are said to be collinear, if there is a line that contains all of them.