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 a1,a2,a3,a4,a5,.... be a GP.
a1 = a and common ratio = r
Then a2 = a × r = ar
a3 = a2 × r = ar × r = ar2
a4 = a3 × r = ar2 × r = ar3
Likewise, an = ran-1
an 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, ar2, ar3, ar4,....
First term = a = a × 1 = a × r0 = a × r1-1
Second term = ar = a × r = a × r1 = a × r2-1 = ar2-1
Third term = ar2 = a × r2 = a × r3-1 = ar3-1
Fourth term = ar3 = a × r3 = a × r4-1 = ar4-1
……………………………………. and so on
nth term = an = arn - 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 = ; ∀ n > 1.
A finite GP → a,ar, ar2, ar3, ar4,....,arn-1
An infinite GP → a, ar ,ar2, ar3,....
Correspondingly, a + ar + ar2 + ar3 + ar4 +....+ arn-1 is called a finite GP series,
whereas a + ar + ar2 + ar3 + ar4 +.... is called an infinite GP series.
Sum of ‘n’ terms of a G.P.
Consider a GP with n terms, a,ar, ar2, ar3, ar4,....,arn-1
a + ar + ar2 + ar3 + ar4 +....+ arn-1 is called a finite GP series.
Sn = a + ar + ar2 + ar3 + ar4 +....+ arn-1 …(i)
While finding the value of Sn, two cases arise.
Case 1. When r = 1, by (i), we have Sn = a + a + a + a + ... + a (n terms) = na
Case 2. When r ≠ 1
Multiplying both the sides of (i) by r, we get
rSn = ar + ar2 + ar3 + ar4 +....+ arn-1 + arn. (ii)
Subtracting (ii) from (i), we have
Sn – rSn = (a + ar + ar2 + ar3 + ar4 +....+ arn-1 ) – (ar + ar2 + ar3 + ar4 +....+ arn-1 + arn)
= a - arn
⇒ (1 - r)Sn = a(1 - rn)
⇒ Sn = ; if r < 1
or ; if r > 1
The general term, tn, of a GP is given by tn = Sn - Sn-1; where Sn denotes the sum of the first n terms and Sn-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/r2, a/r , a, ar, ar2
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 r2 as the common difference.
Thus, four terms are taken as: a/r3, a/r, ar, ar3 and six terms are taken as: a/r5 , a/r3, a/r, ar, ar3, ar5.