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: \(a, ar, ar^2, ar^3…\), where ‘a’ represents the first term and ‘r’ represents the common ratio. The common ratio can 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: \(\frac{1}{2}\), \(\frac{1}{4}\), \(\frac{1}{8}\), \(\frac{1}{16}\), …, \(\frac{1}{32768}\) is a finite geometric progression. Here, \(\frac{1}{32768}\) 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: \(a_1 × a_n = a_2 × a_{ n-1} =…= a_k × a_{n-k+1}\)

• 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 \(GP:  a, ar, ar^2, ar^3\), …
 

• n^th term:
  \(a_n = a × r^{n-1} \space \text{or} \space a_n = r × a_{n-1}\)
   

• Sum of the first n terms:
  \(S_n = a(1 − r^n)/(1 − r), for \space r ≠ 1\)
  \(S_n = n × a\) for r = 1
   

• Sum of infinite terms: 
  \(S_∞ = a / (1 – r)\), when |r| < 1

The sum does not exist when \(|r| ≥ 1\).

Learn how to quickly identify, analyze, and apply geometric progressions in problems and real-life scenarios.

• Identify the common ratio to determine growth or decay.
   
• Apply the n-th term formula.
   
• Memorize finite and infinite sum formulas.
   
• Determine if the series converges or diverges.
   
• Solve real-life problems using GP concepts.

Geometric progression has a vital role in various real-life situations. Let's explore how this concept applies in real-life scenarios.

• Compound Interest in Finance - The amount of money grows exponentially when interest is compounded periodically, forming a GP.
   
• Population Growth - Populations that grow by a fixed percentage over time follow a geometric progression.
   
• Depreciation of Assets - The value of machinery or vehicles often decreases by a fixed ratio annually, modeled using GP.
   
• Radioactive Decay - The remaining quantity of a radioactive substance decreases by a fixed fraction over equal time intervals.
   
• Computer Science & Algorithms - Problems like binary search or tree structures involve exponential growth or halving, which are modeled by geometric progressions.