Geometric Progression

The sequence in which each term is obtained by multiplying the previous term by a fixed number (common ratio) is known as a Geometric Progression. It is usually expressed as: a, ar, ar², ar³…, where ‘a’ represents the first term and ‘r’ represents the common ratio. A common ratio can either be positive or negative. Any term in a GP can be determined using the first term and the common ratio.

Geometric progressions are mainly classified into two types based on their length.

The different types of geometric progressions are:

• Finite geometric progression

• Infinite geometric progression

We will now learn about each type in detail:

Finite geometric progression
A finite geometric progression has a limited number of terms, and the last term is known.

For example: 1/2, 1/4, 1/8, 1/16,…,1/32768 is a finite geometric progression. Here, 1/32768 is the last term.

Infinite geometric progression

An infinite geometric progression has an endless number of terms. Since there is no fixed number of terms, the last term cannot be specified.  

For example, the infinite series 3, −6, +12, −24, +... does not have a definite end term.

To help you identify the sequence effectively, we will now look at the key differences between GP and AP.

Understanding the unique features of a progression helps us identify it more easily. Here are a few properties that geometric progressions (GP) follow. 

• The square of any term in a GP is equal to the product of the terms that are directly adjacent to it: a_k² = a_k-1 × a_k+1

• In a finite geometric progression, terms that are equally spaced from the beginning and the end have the same product: a₁ × a_n = a₂ × a_n-1 =…= a_k × a_n-k+1

• Multiplying or dividing a GP by a non-zero constant, the new sequence remains a GP with the same common ratio.

• The reciprocal of each term in a GP results in another GP with a new common ratio equal to 1/r (r = original common ratio).

• A GP remains a GP even if each of its terms is raised to the same power. 
  Example: GP: a, ar, ar²,...,
  Raise each term to the power k:
  a^k, (ar)^k, (ar²)^k,..., which is still a GP.

In a GP, the sum of the terms can be calculated using the following formulas:

For a GP:  a, ar, ar², ar³, …

n^th term: 

a_n = ar^n–1 or a_n =  r × a_{n– 1}

Sum of the first n terms:

S_n = a(1 − rⁿ)/(1 − r) for r ≠ 1, and

S_n = n × a for r = 1

Sum of infinite terms: 

S_∞ = a/(1 - r), when |r| < 1

The sum does not exist when |r| ≥ 1.

Geometric progression has a vital role in various real-life situations. Let’s now learn how this concept can be applied outside mathematics.

• Geometric progressions help analyze populations that grow at a steady rate, like 20% each year. An example would be 1,000 → 1,200 → 1,440 → 1,728 (20% annual growth).

• GP also describes how a substance decreases by a set proportion over time, such as in radioactive decay.

• The exponential growth of the computer memory units forms a geometric progression. For example: Doubling of memory sizes: 1GB → 2 GB → 4 GB → 8 GB.