In Maths,
Geometric Progression
(GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term past a fixed number, which is chosen a common ratio. This progression is as well known as a
geometric sequence
of numbers that follow a design. Also, learn arithmetic progression hither. The mutual ratio multiplied here to each term to become the next term is a notnil number. An case of a Geometric sequence is two, 4, eight, 16, 32, 64, …, where the common ratio is ii.
 Definition
 Backdrop
 General Grade of GP
 Nth Term
 Common Ratio
 Formula for Sum of Nth Terms of GP
 Types
 Finite Geometric Progression
 Infinite Geometric Progression
 Formula List
 Video lesson
 Solved Examples
 Bug
 FAQs
What is Geometric Sequence?
A geometric progression or a geometric sequence is the sequence, in which each term is varied by some other by a common ratio. The adjacent term of the sequence is produced when we multiply a constant (which is nonzero) to the preceding term. It is represented by:
a, ar, ar^{ii}, ar^{three}, ar^{4}, and so on.
Where a is the first term and r is the mutual ratio.
Notation:
It is to exist noted that when we divide whatsoever succeeding term from its preceding term, so we get the value equal to the common ratio.
Suppose we carve up the 3rd term by the 2nd term nosotros get:
ar^{ii}/ar = r
In the same way:
ar^{3}/ar^{2}
= r
ar^{4}/ar^{iii}
= r
Backdrop of Geometric Progression (GP)
Some of the of import backdrop of GP are listed below:

Three nonzero terms a, b, c are in GP if and only if b^{two}
= airconditioning 
In a GP,
Three consecutive terms can be taken as a/r, a, ar
Four consecutive terms can be taken every bit a/r^{3}
, a/r, ar, ar^{3}
Five consecutive terms can be taken equally a/r^{ii}
, a/r, a, ar, ar^{2}

In a finite GP, the product of the terms equidistant from the beginning and the stop is the same
That means, t_{1}
.t_{n}
= t_{2}
.t_{n1}
= t_{3}
.t_{northward2}
= …..  If each term of a GP is multiplied or divided past a notcipher constant, then the resulting sequence is also a GP with the aforementioned common ratio
 The production and quotient of two GP’s is over again a GP
 If each term of a GP is raised to the power by the same nonzippo quantity, the resultant sequence is too a GP

If a_{i}
, a_{2}
, a_{3}
,… is a GP of positive terms and then log a_{i}
, log a_{two}
, log a_{three}
,… is an AP (arithmetics progression) and vice versa
General Form of Geometric Progression
The general form of Geometric Progression is:
a, ar, ar^{2}, ar^{three}, ar^{4},…, ar^{n1}
Where,
a = First term
r = common ratio
ar^{n1}
= nth term
General Term or Nth Term of Geometric Progression
Allow a be the getgo term and r exist the common ratio for a Geometric Sequence.
Then, the 2d term, a_{2}
= a × r = ar
Third term, a_{3}
= a_{2}
× r = ar × r = ar^{two}
Similarly, nth term, a_{n}
= ar^{n1}
Therefore, the formula to observe the nth term of GP is:
Note: The nth term is the last term of finite GP.
Common Ratio of GP
Consider the sequence a, ar, ar^{2}, ar^{3},……
First term = a
2d term = ar
3rd term = ar^{2}
Similarly, nth term, t_{north}
= ar^{northwardane}
Thus, the common ratio of geometric progression formula is given as:
Common ratio = (Any term) / (Preceding term)
= t_{n}
/ t_{northwardi}
= (ar^{n – one}
) /(ar^{n – 2})
= r
Thus, the full general term of a GP is given by ar^{n1}
and the general form of a GP is a, ar, ar^{2},…..
For Example:
r = t_{2}
/ t_{1}
= ar / a = r
Sum of Northward term of GP
Suppose a, ar, ar^{2}, ar^{three},……ar^{n1}
is the given Geometric Progression.
Then the sum of north terms of GP is given past:
Southward_{northward}
= a + ar + ar^{ii}+ ar^{iii}+…+ ar^{due north1}
The formula to find the sum of n terms of GP is:
S_{n}
= a[(r^{due north}– 1)/(r – 1)] if r ≠ i and r > ane
Where
a is the first term
r is the mutual ratio
north is the number of terms
As well, if the common ratio is equal to i, then the sum of the GP is given by:
Types of Geometric Progression
Geometric progression can be divided into two types based on the number of terms information technology has. They are:
 Finite geometric progression (Finite GP)
 Infinite geometric progression (Infinite GP)
These ii GPs are explained below with their representations and the formulas to find the sum.
Finite Geometric Progression
The terms of a finite One thousand.P. can be written equally a, ar, ar^{2}, ar^{3},……ar^{nane}
a, ar, ar^{2}, ar^{3},……ar^{nane}
is chosen finite geometric series.
The sum of finite Geometric series is given by:
S_{n}
= a[(r^{due north}– ane)/(r – 1)] if r ≠ 1 and r > 1
Infinite Geometric Progression
Terms of an infinite Thou.P. can be written as a, ar, ar^{2}, ar^{iii}, ……ar^{none},…….
a, ar, ar^{2}, ar^{3}, ……ar^{nane},……. is called infinite geometric series.
The sum of infinite geometric serial is given past:
\(\begin{assortment}{l}\sum_{k=0}^{\infty}\left(a r^{k}\right)=a\left(\frac{1}{1r}\right)\end{assortment} \)
This is chosen the geometric progression formula of sum to infinity.
Geometric Progression Formulas
The listing of formulas related to GP is given below which will assistance in solving different types of bug.

The general form of terms of a GP is a, ar, ar
^{2}
, ar
^{3}
, and so on. Hither, a is the first term and r is the mutual ratio. 
The nth term of a GP is T
_{north}
= ar^{n1}

Common ratio = r = T_{n}
/ T_{nane}

The formula to summate the sum of the start northward terms of a GP is given by:
S_{n}
= a[(r^{n}
– 1)/(r – 1)] if r ≠ 1and r > 1
S_{n}
= a[(1 – r^{n}
)/(1 – r)] if r ≠ one and r < 1  The nth term from the end of the GP with the final term 50 and common ratio r = fifty/ [r(n – 1)].

The sum of infinite, i.e. the sum of a GP with infinite terms is S
_{∞}
= a/(1 – r) such that 0 < r < 1. 
If three quantities are in GP, then the eye one is called the
geometric mean
of the other two terms. 
If a, b and c are three quantities in GP, so and b is the geometric mean of a and c. This can be written every bit
b^{2}
= ac
or
b =√ac  Suppose a and r be the kickoff term and common ratio respectively of a finite GP with due north terms. Thus, the kth term from the end of the GP volition be = ar
^{nk}
.
Video Lesson
Applied Concept – Sum of Infinite Terms of G.P.
Solved Examples of Geometric Progression
Question 1: If the first term is 10 and the common ratio of a GP is three, then write the showtime 5 terms of GP.
Solution: Given,
Start term, a = 10
Common ratio, r = 3
We know the general form of GP for first 5 terms is given past:
a, ar, ar^{2}, ar^{3}, ar^{4}
a = 10
ar = 10
×
3 = xxx
ar^{2}
= 10×
3^{ii}
= 10×
9 = xc
ar^{3}
= 10×
3^{3}
= 270
ar^{4}
= 10×
iii^{4}
= 810
Therefore, the starting time five terms of GP with 10 as the first term and 3 as the common ratio are:
10, thirty, 90, 270 and 810
Question 2: Find the sum of GP: x, 30, 90, 270 and 810, using formula.
Solution: Given GP is 10, 30, 90, 270 and 810
Starting time term, a = 10
Mutual ratio, r = 30/x = 3 > 1
Number of terms, n = v
Sum of GP is given past;
South_{n}
= a[(r^{n}– ane)/(r – 1)]
S_{5}
= ten[(3^{5}– 1)/(3 – one)]
= x[(243 – ane)/2]
= 10[242/ii]
= 10
×
121
= 1210
Bank check: 10 + 30 + 90 + 270 + 810 = 1210
Question 3: If 2, 4, 8,…., is the GP, then find its 10th term.
Solution: The nth term of GP is given by:
ii, 4, 8,….
Here, a = 2 and r = 4/2 = 2
a_{n}
= ar^{northward1}
Therefore,
a_{x}
= 2 x 2^{ten – 1}
= 2×
two^{nine}
= 1024
Exercise Problems on Geometric Progression
 Find the equivalent fraction of the recurring decimal 0.595959…..
 What is the 12th term of the sequence 4, eight, 16, 64,….?

Bank check whether the given sequence is GP.
27, 9, 3, …  Write the commencement five terms of a GP whose first term is 3 and the mutual ratio is 2.
Frequently Asked Questions on Geometric Progression
What is a Geometric Progression?
Geometric Progression (GP) is a blazon of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio.
Give an example of Geometric Progression.
The example of GP is: 3, halfdozen, 12, 24, 48, 96,…
What is the general class of GP?
The general form of a Geometric Progression (GP) is given by a, ar, ar^{2}, ar^{3}, ar^{4},…,ar^{none}
a = Beginning term
r = mutual ratio
ar^{north1}
= nth term
What is the mutual ratio in GP?
The common multiple betwixt each successive term and preceding term in a GP is the mutual ratio. It is a abiding value that is multiplied past each term to get the side by side term in the Geometric serial. If a is the first term and ar is the side by side term, and then the mutual ratio is equal to:
ar/a = r
What is not a geometric progression?
If the mutual ratio between each term of a geometric progression is non equal so it is non a GP.
What is the sum of a geometric series?
If a, ar, ar^{2}, ar^{3},……ar^{nane}
is the given Geometric Progression, then the formula to find sum of GP is:
S_{n}
= a + ar + ar^{ii}+ ar^{3}+…+ ar^{n1}
Or
Southward_{n}= a[(r^{n}– 1)/(r – 1)] where r ≠ 1 and r > ane
What is the Common Ratio of the Geometric Sequence Below
Source: https://byjus.com/maths/geometricprogression/
Originally posted 20220804 18:09:32.