Geometric Progression : Basics

Hemant Pandey
30/05/2018 0 0

Geometric Mean (GM)


The geometric mean is similar to the arithmetic mean, but it uses the property of multiplication, to find the mean between any two numbers.

GM of two numbers a and b may be defined as G = √(ab)

For example, GM of 2 and 3 is √6.

We can find a general formula for inserting a GM between any two arbitrary numbers a and b.


 If G is the geometric mean between a and b, then by definition of GP a,  G,  b are in GP.

Or G2=a.b


We can also find a formula for inserting n GM in between two numbers a and b.

Let  G1,  G2,  G3  ….  Gn be then GMs between a and b.

Now a,   G1,  G2, G3…. Gn,  b are in GP.

By definition of the last term of a GP  b = (n+2)th term of GP.

If R=common ratio, then by definition of GP,  b = a.Rn+1      

Now G1=a?R



   and Gn=a?Rn


We will take a few examples of GM too.



Find the GM of the following and insert n geometric means between them.


    e and 1
    7 and tan x
    log x and sec-1x



(i) The GM between e and 1 is e. To insert n GMs, we use the definition of geometric progressions.

Let  G1,   G2,  G3,  …  Gn,  be n  geometric means inserted between e and 1.

Also, let common ratio be R.

 By the definition of GP    = (n+2)th term of the GP.

 1 =eRn+1

Or R= (1/e)1/(n+1)

Hence   G1 = e⋅R = e⋅(1/e)1/(n+1)



            and  Gn=e⋅R=e⋅(1/e)n/(n+1)



(ii)   Proceeding as before the GM  of 7 and tanx is √(7tanx).

Let G1, G2, G3,… Gn,  be n  geometric means inserted between 7 and tanx.Also, let common ratio be R.

To get the n geometric means, we proceed as before.

Here tanx=(n+2)th term.


Or Rn+1= tanx/7

R = (tanx/7)1/n+1

 G1 = 7R = 7⋅(tanx/7)1/n+1

    G2  =  7R2   = 7⋅(tanx/7)2/n+1

    G3 = 7R3 = 7⋅(tanx/7)3/n+1


and   Gn=7Rn=7(tanx/7)n/n+1


(iii)  Proceeding exactly as before, the GM of logx and sec-1x will be √(logx⋅sec-1x)

 Let   G1,  G2,  G3, …  Gn,  be n  geometric means inserted between logx and sec-1x.

Also, let common ratio be R.


To insert n GMs between logx and sec-1x, we take the sequence as

    logx, G1, G2, G3,… Gn, sec-1x.

 Gn = (n+2)th term =  sec-1x = logx.Rn+1

 R= (sec-1x/logx)1/(n+1)

Now as before G1=logx⋅R=logx⋅(sec-1x/logx)1/(n+1)



           and Gn=logx⋅Rn=logx⋅(sec-1x/logx)n/(n+1)

0 Dislike
Follow 2

Please Enter a comment


Other Lessons for You

Geometric Progression
If a is the first term and r is the common ratio of the geometric progression, then its nth term is given by an = arn-1 The sum Sn of the first n terms of the G.P. is given by Sn = a (rn – 1)/...

Raj Kumar | 06/09/2018

0 0

Notes on Harmonic progression
Let a, b and c form an H.P. Then 1/a, 1/b and 1/c form an A.P. If a, b and c are in H.P. then 2/b = 1/a + 1/c, which can be simplified as b = 2ac/(a+c) If a and b are two non-zero numbers...

Raj Kumar | 23/08/2018

0 0

copycat syndrome
To be good at something, it is better to copy. Most projects are now on google and reports are just copy pasted into a book. Duplication is a mantra in most marketing firms and some copy to the T. Effective...

Suraj Mohan | 22/10/2016

0 0

Sequences : An Overview
Sequences Mathematics is the science of patterns and homogeneity. Sequences are all about them. A sequence has a traditional meaning in mathematics similar to that in ordinary language. A sequence is...

Hemant Pandey | 30/05/2018

0 0

Find Tutors, Trainers & Institutes near you

Post requirement and connect with the tutors in your locality

  • Select the best Tutor
  • Book & Attend a Free Demo
  • Pay and start Learning

Want to learn something New?

Find best tutors, trainers & institutes near you on UrbanPro

Post your learning requirement

UrbanPro.com is India's largest network of most trusted tutors and institutes. Over 55 lakh students rely on UrbanPro.com, to fulfill their learning requirements across 1,000+ categories. Using UrbanPro.com, parents, and students can compare multiple Tutors and Institutes and choose the one that best suits their requirements. More than 7.5 lakh verified Tutors and Institutes are helping millions of students every day and growing their tutoring business on UrbanPro.com. Whether you are looking for a tutor to learn mathematics, a German language trainer to brush up your German language skills or an institute to upgrade your IT skills, we have got the best selection of Tutors and Training Institutes for you. Read more