Geometric Progression

Geometric progression is a type of progression where every subsequent term has a constant ratio with its preceding term. It is a special kind of sequence in which to find the subsequent term; we have to multiply with a fixed term referred to as the common ratio. Similarly, to find the preceding term in the progression, we divide it with the common ratio. A geometric progression can be both a finite and an infinite sequence. The common ratio in the sequence can be negative or positive. Some geometric sequence examples are 2,4,8,16,32,64…..where the common ratio is 2 and 10, 30, 90, 270 …. Where the common ratio is 3.  

Types of Geometric Progression

 Geometric progression, also referred to as GP, is represented through a sequence a, ar, ar²….. where a stands for the first term, and r stands for the common ratio. To find the geometric progression sequence, all we need to know is the first term and the common ratio. 

There are 2 types of geometric progression based on the number of terms present, namely finite geometric progression and infinite geometric progression. 

1. Finite Geometric Progression: Finite geometric progression has a finite number of terms, where the last term is definite and known. For example, 2,4,8,16,32,64….512 is a finite GP where the last term is 512. 
2. Infinite Geometric Progression: Infinite geometric progression has infinite terms where the last term is not defined. For example: 2,4,8,16,32,64….where the last term is not defined. 

Geometric Progression Formula

The geometric sequence formula identifies the nth term in the sequence. All we need is the common ratio and the first term in the sequence. If the common ratio is unknown, we can find it by deriving the ratio of any term with its preceding term.

The formula for finding the nth term in a GP sequence is 

an = arn-1

Where a is the first term and r is the common ratio

Further, a geometric progression sum formula is also used to derive the sum of all the terms in a GP. Since there are two kinds of GP series, there are 2 formulas to calculate the sum of geometric progression. 

• Sum of finite geometric progression: 

Sn = a(1−rn)/(1−r) for r≠1

and 

Sn = an for r = 1

Where a is the first term 

r is the common ratio and n is the number of terms in the series

• Sum of Infinite geometric progression: 

When |r| < 1 

S∞ = a/(1 – r)

Where a is the first term

r is the common ratio

And

When |r| > 1 

There is no sum since the series does not converge. The sum tends to infinity in this case and cannot be defined.

Properties of Geometric Progression, 

Some interesting properties of geometric progression that will help you understand the concept better are: 

• Three non-zero terms, a, b, and c, can be in a GP sequence only if b² = ac
• In a geometric progression, 

           3 consecutive terms can be represented as a/r, a, ar

           4 consecutive terms can be represented as a/r³, a/r, ar, ar³

           5 consecutive terms can be represented as a/r², a/r, a, ar, ar²

This method of representation in certain cases makes our calculations easier.

•  In a finite Geometric progression, the product of the terms at equal distance from the beginning and the end is the same. This means t1.tn = t2.tn-1 = t3.tn-2 = …..
• Suppose every term in a GP series is multiplied or divided by a non-zero constant. In that case, the resulting sequence will also be a geometric progression having the same common ratio.
• The product or quotient of 2 geometric progression series is also a GP series.
• If every term of a geometric progression series is raised to the power by a similar non-zero quantity, the resulting sequence will also be a GP sequence.
• If a1, a2, a3,… represent the geometric progression of positive numbers then, log a1, log a2, log a3,… will be an arithmetic series and vice versa.  

The geometric mean, b of two terms a and c can be derived by √(ac). 

Proof:If the terms a, b, and c are in a geometric progression sequence, then the ratio of the consecutive terms will be the same.

b/a = c/b

Or, b² = ac

b = √(ac)

Thus, the geometric mean of a and c is b. 

Further, there can also be multiple geometric means. If a and b are the two GP numbers, let G1, G2, G3….Gn is n geometric mean between them. 

In this case, Gm = a(b/a)m/n+1

For example, If 1 and 256 are 2 numbers between which 3 more numbers are to be inserted so that the resulting sequence is a geometric progression, we can assume the numbers as G1, G2, G3

The sequence will be – 1, G1, G2, G3,256

We are aware that the general form of any GP sequence is a1, a1r, a1r², a1r³, a1rn-1, a1rn…

Here, a1 = 1 and ar4 = 256

or, r4 = (4)4

or, r = 4, -4

Our GP series is 1 ar ar² ar³ 256

Case I: 

When r=4, 

ar = 1 × 4

ar² = 1 × 4²

ar³ = 1 × 4³

GP: 1, 4, 16, 64, 256

Here, 4,16, and 64 are the three numbers that can be added between 1 and 256 so that the series is a GP sequence. 

Case II:

When r=(-4), 

ar = 1 × -4

ar² = 1 × -4²

ar³ = 1 ×-4³

GP: 1, -4, 16, -64, 256

Here, -4, 16, and -64 are the three numbers that can be added between 1 and 256 so that the series is a GP sequence. 

Difference Between Geometric Progression and Arithmetic Progression