. Geometric Progression In the diagram given in Fig. . , D DEF is formed by joining the mid points of the sides AB, BC and CA of D ABC .
Then the size of the triangle D DEF is exactly one-fourth of the size of D ABC . Similarly D GHI is also one-fourth of D DEF and so on. In general, the successive areas are one-fourth of the previous areas. The area of these triangles are D ABC , D ABC, ´ D ABC ,...
That is, D ABC , D ABC , D ABC ,... In this case, we see that beginning with D ABC , we see that the successive triangles are formed whose areas are precisely one-fourth the area of the previous triangle. So, each term is obtained by multiplying to the previous term. As another case, let us consider that a viral disease is spreading in a way such that at any stage two new persons get affected from an affected person.
At first stage, one person is affected, at second stage two persons are affected and is spreading to four persons and so on. Then, number of persons affected at each stage are , , , , ... where except the first term, each term is precisely twice the previous term. From the above examples, it is clear that each term is got by multiplying a fixed number to the preceding number.
This idea leads us to the concept of Geometric Progression. Definition A Geometric Progression is a sequence in which each term is obtained by multiplying a fixed non-zero number to the preceding term except the first term. The fixed number is called common ratio. The common ratio is usually denoted by r .
. . General form of Geometric Progression Let a and r ¹ be real numbers. Then the numbers of the form a ar ar ar n , ...
... is called a Geometric Progression. The number ‘ a ’ is called the first term and number ‘ r ’ is called the common ratio. We note that beginning with first term a , each term is obtained by multiplied with the common ratio ‘ r ’ to give ar ar ar ,...
. . General term of Geometric Progression We try to find a formula for n th term or general term of Geometric Progression (G.P.) whose terms are in the common ratio. K H E F I L J G D Fig.
. Fig. . Numbers and Sequences a ar ar ar n ,..., ,...
where a is the first term and ‘ r ’ is the common ratio. Let t n be the n th term of the G.P. Then t = r = r t = t t = t ar ar ar t n = −+ t ar ar ar Thus, the general term or n th term of a G.P. is t n = ar n -