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Last Updated: Mar 27, 2024

Geometric Progression(G.P.)

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23 Jul, 2024 @ 01:30 PM

Introduction

Progression is defined as a series that exhibits a specific pattern. It is also known as a sequence. Every (n+1)th term in a progression is obtained by applying a specific rule on its nth term. For example, 10, 20, 30 ... is a progression because every number is obtained by adding 10 to its previous number. This pattern may vary from progression to progression.

In mathematics, there are mainly three types of progressions: 

  1. Arithmetic progressions
  2. Harmonic progressions
  3. Geometric progressions
     

This blog will discuss one of the types of progression,i.e., Geometric Progression.

Geometric Progression

The term Geometric progression(G.P.) occurs in the topic sequence and series. G.P. is a sequence such that any element after the first is obtained by multiplying the previous element by a constant factor. 

For example: 

Assume we have a sequence 3, 6, 12, 24,...

Here we can see that the constant factor is 2. Every element after the first element is obtained by multiplying it with 2, by which we get the sequence like 3, 3*2=6, 6*2=12, 12*2=24, 24*2=48, and so on.
 

By calculating the common ratio, which is denoted by r, we can determine if a sequence is in Geometric Progression(G.P.). The common ratio (r) can be calculated by dividing any term by the previous term.

Let terms be,

T1 = 3

T2 = 6 

T3 = 12

T4 =24

Therefore, 

r = Term nTern n-1 => if their common ratio are equal it means they are in G.P. 

T2T1 = T3T2 => 63 = 126 => 2 = 2, hence it is in G.P.

General Term

Let’s see how we can derive the general formula for G.P.

Consider the above sequence i.e., 3, 6, 12, 24,...

Here, 3 = a = represented as the first term.

2 = r = represented as the common ratio.

So the sequence can be written as: 3, 3*2, 3*2*2, 3*2*2*2,... 

Therefore general term of G.P. is  a, ar, ar2, ar3,... (where r is the common ratio which is 2).

How to find the nth term of Geometric Progression(G.P.)?

We know that the general term is: a, ar, ar2, ar3,...

T1 = 3 = a

T2 = 6 = ar

T3 = 12 = ar2

T4 =24 = ar3

We can see here that the first term is constant, but the power of the common ratio r is increasing hence the nth term, i.e., the last term = l = Tn = a*rn-1.

So the last term and 18 th term for the sequence above will be

l = Tn = 3*2n-1.

and T18 = 3*218-1 = 3*217

How to find the sum of the nth term in Geometric Progression(G.P.)?

We know that the general term is a, ar, ar2, ar3,...

Therefore the sum will be:

S = a + ar + ar+ ar+ … + arn-1--------------- eq(i)

Now multiplying both side with r, we get

rS = ar + ar+ ar+ar4 … + arn-1+arn-------------eq(ii)

Subtract eq(i)from eq(ii), we get

rS-S = (ar + ar+ ar+ar4 … + arn-1+arn)-(a + ar + ar+ ar+ … + arn-1)

Hence,

S(r-1) = arn-a

S = a(rn-1) / (r-1)   (when r >1)

and

S = a(1-rn) / (1-r)   (when r <1)

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Properties of Geometric Progression (G.P.)

  • Generally, three non-zero terms, a, b, and c, are in G.P. if and only if 2b = Ac.
  • You can take three consecutive terms as a/r, a, and ar.
  • You can accept four consecutive terms as a/r3, a/r, ar, ar3.
  • You can take five consecutive terms as a/r2, a/r, a, ar, ar2.
     

A finite G.P. produces the same product when terms are equidistant from the beginning and the end.

  • Therefore, T1.Tn = T2.Tn-1 = T3.Tn-2 = .....
  • A GP with a consistent common ratio will also result from multiplying or dividing each term with a non-zero constant.
  • When two G.P.s are multiplied or divided, they become a G.P. again.
  • Any term of a G.P. raised to the power of a non-zero quantity will also be a G.P.

Types of Geometric Progression(G.P.)

  1. Finite Geometric Progression (G.P.): Finite Geometric Progression (G.P.): The terms of a finite G.P. can be written as a, ar, ar2, ar3,……arn-1. In this, the last term is known. 
    Sum of Finite G.P. is:
    Sn = a[(rn-1)/(r-1)] if r ≠ 1
     
  2. Infinite Geometric Progression(G.P.): The Terms of an infinite G.P. can be written as a, ar, ar2, ar3, ……arn-1,…….In this, the last term is unknown.
    Sum of infinite (G.P.) is:
    S∞ = a/(1 - r) for |r| < 1

Check out this problem - Shortest Common Supersequence.

FAQs

  1. Difference Between Arithmetic and Geometric Progression?
    An arithmetic progression (A.P.) is characterized by each successive term being less than the previous one by a given number. The geometric progression is one in which each consecutive term is the product of the previous one by a constant number. 
     
  2. What is not a geometric progression?
    Geometric progressions are not G.P.s if the ratio between each term is not equal.  
     
  3. Is it possible for zero to be a part of a geometric series?
    There will be no geometric progression if the first term is 0. 
     
  4. Is there an infinite geometric progression without a finite sum?
    With a common ratio r > 1, the infinite geometric series cannot have a finite sum.

Key Takeaways

In this blog, we have seen what Geometric progression(G.P.) is and how to find the general term, nth term, and the sum of nth terms of a G.P.

Recommended Readings:

We hope that this blog has helped you enhance your knowledge about Geometric Progression(G.P.) and if you would like to learn more, check out our articles on the link. Do upvote our blog to help other ninjas grow. Happy Coding!

Topics covered
1.
Introduction
2.
Geometric Progression
2.1.
General Term
2.2.
How to find the nth term of Geometric Progression(G.P.)?
2.3.
How to find the sum of the nth term in Geometric Progression(G.P.)?
3.
Properties of Geometric Progression (G.P.)
4.
Types of Geometric Progression(G.P.)
5.
FAQs
6.
Key Takeaways