Geometric progression

About geometric progression

A G.P. is a sequence whose first term is non-zero and each of whose succeeding term is r times the preceding term, where r is some fixed non - zero number, known as the common ratio of the G.P. For example 3 + 9 + 27 + 81 is a G.P. whose common ratio is 3.Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah.

n^th Term and Sum of n Terms:

If a is the first term and r the common ratio, then G.P. can be written as a, ar, ar², . . . the nth term, an, is given by an = ar^n-1. The sum S_n of the first n terms of the G.P. is

If -1 < r < 1, then the sum of the infinite G.P. a + ar + ar² +........=

Important points about geometric progression

If each term of a G.P. is multiplied (divided) by a fixed non-zero constant, then the resulting sequence is also a G.P. with same ratio as that of the given G.P. .
If each term of a G.P. (with common ratio r) is raised to the power k, then the resulting sequence is also a G.P. with common ratio rk.
If a₁, a₂, a₃, .... and b₁, b₂, b₃, .... are two G.P.’s with common ratios r and r′ respectively then the sequence a₁b₁ , a₂b₂, a₃b₃.....is also a G.P. with common ratio r r′.
If we have to take three terms in a G.P., it is convenient to take them as a/r, a, ar. In general, we take in case we have to take (2k + 1) terms in
a G.P.
If we have to take four terms in a G.P., it is convenient to take them as a/r3, a/r, ar, ar3 . In general, we take , in case we have to take 2k terms in a G.P.
If a₁, a₂, …., an are in G.P., then a₁a_n = a₂ a_n-1 = a₃a_n-2 = . . . .
If a₁, a₂, a₃,.…is a G.P. (each aI > 0), then loga₁, loga₂, loga₃ ….. is an A.P. The converse is also true.

Geometric Means

If three terms are in G.P., then the middle term is called the geometric mean (G.M.) between the two. So if a, b, c are in G.P. then b = is the geometric mean of a and c.

If a₁, a₂......an are non-zero positive numbers then their G.M (G) is given by
G = (a₁a₂a₃......an)^1/n. If G₁, G₂,…..Gn are n geometric means between a and b then
a, G₁, G₂, …., G_n, b will be a G.P. Here b = a r^{n + 1}