Geometric progression (G.P.):
A G.P. is a sequence whose first expression is non-zero and each of whose next term is r times the previous term, where r is some fixed non - zero number, Called as the common ratio of the G.P. For example 3 + 9 + 27 + 81 is a G.P. whose ratio is 3.
nth Term and addition of n Terms:
If a is the first term of g.p. and r the common ratio, then G.P. may be defined as a, ar, ar2, . . . the nth term, an, is provided by an = arn-1. The sum Sn of the first n terms of the G.P. is
![378_Geometric progression.png](https://www.expertsmind.com/../CMSImages/378_Geometric progression.png)
If -1 < r < 1, then the summation of the infinite G.P. a + ar + ar2 +........=a/r-1
Notes
- If each and every term of a G.P. is divided or multiplied by a fixed non-zero constant, then the resultant sequence is also a G.P. with similar ratio as that of the specified G.P. .
- If each term of a G.P. (with common ratio r) is gained to the power k, then the derived sequence is also a G.P. with common ratio rk.
- If a1, a2, a3, .... and b1, b2, b3, .... are two G.P.'s with basic ratios r and r¢ respectively then the progression a1b1 , a2b2, a3b3.....is also a G.P. with common ratio r r¢.
- If we have to put three terms in a G.P., it is convenient to define them as a/r, a, ar. In general, we have to take
in that case we have to take (2k +1) terms in a G.P.
- If we need to take four terms in a G.P., it is convenient to take them as a/r3, a/r, ar, ar3 . Usually we take
, in that case we have to take 2k parts in a G.P.
- If a1, a2, ...., an are in G.P., then a1an = a2 an-1 = a3 an-2 = . . . .
- If a1, a2, a3,....is a G.P. (each aI > 0), then loga1, loga2, loga3 ..... is an A.P. The reverse is also true.
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