Geometric Progression(G.P.)

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 n^th 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*r^n-1.

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

l = Tn = 3*2^n-1.

and T₁₈ = 3*2^18-1 = 3*2¹⁷

How to find the sum of the n^th 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 + ar² + ar³ + … + ar^n-1--------------- eq(i)

Now multiplying both side with r, we get

rS = ar + ar² + ar³ +ar⁴ … + ar^n-1+arⁿ-------------eq(ii)

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

rS-S = (ar + ar² + ar³ +ar⁴ … + ar^n-1+arⁿ)-(a + ar + ar² + ar³ + … + ar^n-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)