Harmonic Progression (H.P.)
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.
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.)
The combination of arithmetic and geometric progression is called arithmetico–geometric progression.
nth term of A.G.P.
If a1, a2, a3, a4,.....an is an A.P.
b1, b2, b3....bn is a G.P.,
then the sequence a1b1, a2b2, a3b3,.......anbn 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.Read More...
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.
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.
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.
Three or more points are said to be collinear, if there is a line that contains all of them.Read More...