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Geometric Progressions

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Geometric Progressions - Lesson Summary

A sequence is said to be a Geometric Progression (GP), if each term (except the first) is obtained by multiplying its preceding term by a non-zero constant. 

This constant is called the common ratio and is obtained by dividing any term (except the first term) by its preceding term. It is denoted by r.

Let a 1,a 2,a 3,a 4,a 5,.... be a GP.

a 1 = a and common ratio = r

Then a 2 = a × r = ar

a 3 = a 2 × r = ar × r = ar 2

a 4 = a 3 × r = ar 2 × r = ar 3

Likewise, a n = ra n-1

a n is called the general term of the GP.
 
Any required term in the sequence can be found without actually finding the preceding terms.
 
Thus, the general form of a GP is  a,ar, ar 2, ar 3, ar 4,....

First term = a = a × 1 = a × r 0 = a × r 1-1

Second term = ar = a × r = a × r 1 = a × r 2-1 = ar 2-1

Third term = ar 2 = a × r 2 = a × r 3-1 = ar 3-1

Fourth term = ar 3 = a × r 3 = a × r 4-1 = ar 4-1

……………………………………. and so on

n th term = a n = ar n - 1

Here, the common ratio, ‘r,’ is defined as the ratio of any term (except the first) to its preceding term,

and is expressed as r =  a r n - 1 a r n - 2 ; ∀ n > 1.

A finite GP →   a,ar, ar 2, ar 3, ar 4,....,ar n-1

An infinite GP →   a, ar ,ar 2, ar 3,....

Correspondingly, a + ar + ar 2 + ar 3 + ar 4 +....+ ar n-1  is called a finite GP series,
whereas a + ar + ar 2 + ar 3 + ar 4 +.... is called an infinite GP series.
 
Sum of ‘n’ terms of a G.P.
 
Consider a GP with n terms, a,ar, ar 2, ar 3, ar 4,....,ar n-1

a + ar + ar 2 + ar 3 + ar 4 +....+ ar n-1  is called a finite GP series.

S n = a + ar + ar 2 + ar 3 + ar 4 +....+ ar n-1    …(i)
 
While finding the value of S n, two cases arise.

Case 1. When r = 1, by (i), we have S n = a + a + a + a + ... + a (n terms) = na

Case 2. When r ≠ 1

Multiplying both the sides of (i) by r, we get
rS n = ar + ar 2 + ar 3 + ar 4 +....+ ar n-1 + ar n. (ii)

Subtracting (ii) from (i), we have
S n – rS n = (a + ar + ar 2 + ar 3 + ar 4 +....+ ar n-1 ) – (ar + ar 2 + ar 3 + ar 4 +....+ ar n-1 + ar n)

             = a - ar n

⇒ (1 - r)S n = a(1 - r n)

⇒ S n =  a(1 - r n ) 1 - r ; if r < 1


or    a(1 - r n ) 1 - r  ; if r > 1


The general term, t n, of a GP is given by t n = S n - S n-1; where S n denotes the sum of the first n terms and S n-1 denotes the sum of the first (n-1) terms of the GP.
 
Simple tricks to find the terms of a GP when their sum is given:
 
When we have to find an odd number of terms in a GP whose sum is given, then it is convenient to take the middle term as a and the common ratio as r.
 
Thus, three terms are taken as: a/r, a, ar and five terms are taken as a/r 2, a/r , a, ar, ar 2
When we have to find an even number of terms in a GP whose sum is given, we take a/r and ar as the two middle terms and r 2 as the common difference.

Thus, four terms are taken as: a/r 3, a/r, ar, ar 3 and six terms are taken as:  a/r 5 , a/r 3, a/r, ar, ar 3, ar 5.

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