# Geometric Progression (GP)

If

*a*is the first term and*r*is the common ratio, then GP can be written as*a*,*ar*,*ar*^{2},*ar*^{3},*ar*^{4}, …,*ar*^{n}^{–1}.

*n***th term:**

*T*=

_{n}*ar*

^{n}^{–1}=

*l*(last term),

where

*r*= ,*n*≥ 2.

*n***th term from last:**

*T*′ = .

_{n}# Sum of *n* terms of a GP

The sum of

*n*terms of a GP with first term “*a*” and common ratio “*r*” is given by*S*=

_{n}*a*or

*S*=

_{n}*a*,

*r*≠ 1

# Some important facts about GP

- If each term of a GP is multiplied or divided by some fixed non-zero number, then the resulting sequence is also a GP.
- If
*x*_{1},*x*_{2},*x*_{3}, ..., and*y*_{1},*y*_{2},*y*_{3}, …, are two GP’s then*x*_{1}*y*_{1},*x*_{2}*y*_{2},*x*_{3}*y*_{3}, …, and , are also GP’s. - If
*x*_{1},*x*_{2},*x*_{3}, … is a GP of positive terms then log*x*_{1}, log*x*_{2}, log*x*_{3}, … is an AP and vice versa. - Three numbers in GP can be taken as
*a*/*r*,*a*,*ar*and four terms in GP as*a*/*r*^{3}*a*/*r*,*ar*,*ar*^{3}. This presentation is useful if product of terms is involved in the problem, otherwise terms should be taken as*a*,*ar*,*ar*^{2}…. - In GP,
*a*_{1},*a*_{2},*a*_{3},…,*a*_{n}_{–1},*a*:_{n }*a*_{1}*a*=_{n}*a*_{2}*a*_{ n–1}=*a*_{3}*a*_{n}_{–2}= … or product of equidistant terms from start and end is same.

# Insertion of Geometric Means

Let

*a*and*b*be two given numbers. If*n*numbers*G*_{1},*G*_{2}, …,*G*are inserted between_{n}*a*and*b*such that the sequence*a*,*G*_{1},*G*_{2}, …,*G*,_{n}*b*is a GP. Then the numbers*G*_{1},*G*_{2}, …,*G*are known as_{n}*n*geometric means (GM’s) between*a*and*b*.

*Notes:**p*th mean =*G*=_{p}*ar*=^{p}*a*.- If
*n*geometric means are inserted between two numbers, then the product of*n*geometric means is the*n*th power of the single geometric mean between the two numbers.*G*_{1}*G*_{2}*G*_{3}…*G*= ()_{n}=^{n}*G*^{n}*G*= is the single geometric mean between*a*and*b*.