# Formulas For Geometric Progression

## Definition and formulas of Geometric Progression

Geometric Progression (GP) is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant. The common ratio (r) is obtained by dividing any term by the preceding term, i.e., r =  a2/a1=a3/a2  = …..an/a{n-1} . Each number in the sequence is known as ‘term’ and the first term of the sequence is called ‘first term’ which is denoted by ‘a’.

 General form of an GP a, ar, ar2, ar3, ……. First Term a Common Ratio r

#### For example:

Given sequence or series is 1, 3, 9, 27, 81…..

Here, a = 1 and r = 3 ( 3/1 ,9/3 , ….) ### Formulas for GP

• Common ratio
r = a2/a1
• nth term of an GP

an = a1rn-1
where, r = common ratio, a1= first term, an-1= the term before the nth term, n = number of terms

• Sum of first n terms in an GP

Sn = (a1r) /(r-1) (if r > 1)

and

a1(1-rn)/1-r (if r < 1)
where, r = common ratio, a1= first term , n = number of terms

• Sum of an infinite GP

S¥ = a/1-r (if  -1<r<1)

• Geometric Mean (GM)

If two non-zero numbers a and b are in GP, then there GM is

GM = √ab

### Properties of Geometric Progression

• If ‘a’ is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be = arm-n.
• The nth term from the end of the G.P. with the last term ‘l’ and common ratio r is   l/(r(n-1))
• Reciprocal of all the term in G.P are also considered in the form of G.P.
• When all terms is GP raised to same power, the new series of geometric progression is form.