Geometric progression

Geometric Progression

A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This is in contrast to an Arithmetic progression, where the difference between consecutive terms is constant. Geometric progressions are fundamental in many areas of mathematics, including Algebra, Calculus, and have applications in finance, computing, and even biology. Understanding them is crucial for anyone venturing into quantitative fields, and surprisingly, even in understanding certain aspects of Technical Analysis in financial markets.

Defining a Geometric Progression

A geometric progression is a sequence of the form:

a, ar, ar², ar³, ..., ar^n-1

where:

'a' is the first term of the sequence.
'r' is the common ratio.
'n' is the number of terms in the sequence.

For example:

2, 6, 18, 54, ... is a geometric progression with a = 2 and r = 3.
100, 50, 25, 12.5, ... is a geometric progression with a = 100 and r = 0.5.
-1, 2, -4, 8, ... is a geometric progression with a = -1 and r = -2.

Notice that to get from one term to the next, you either multiply or divide by the common ratio. In the first example, you multiply by 3. In the second, you multiply by 0.5. In the third, you multiply by -2.

The Formula for the nth Term

The 'n'th term (a_n) of a geometric progression can be calculated using the following formula:

a_n = a * r^(n-1)

Where:

a_n is the nth term
a is the first term
r is the common ratio
n is the term number

Let's illustrate with an example. Suppose we have a geometric progression with a = 5 and r = 2. We want to find the 6th term (a₆).

a₆ = 5 * 2^(6-1) = 5 * 2⁵ = 5 * 32 = 160

Therefore, the 6th term of the sequence is 160. This formula is essential for predicting future values within a geometric progression, which is directly applicable to understanding compound interest and exponential growth – concepts frequently used in Financial Modeling.

The Sum of a Finite Geometric Series

A geometric series is the sum of the terms of a geometric progression. The formula for the sum (S_n) of the first 'n' terms of a geometric series is:

S_n = a * (1 - rⁿ) / (1 - r) (where r ≠ 1)

If r = 1, the sum is simply S_n = n * a.

Let's consider the geometric progression 3, 6, 12, 24, 48. We want to find the sum of the first 5 terms. Here, a = 3, r = 2, and n = 5.

S₅ = 3 * (1 - 2⁵) / (1 - 2) = 3 * (1 - 32) / (-1) = 3 * (-31) / (-1) = 3 * 31 = 93

So, the sum of the first 5 terms is 93. This formula is frequently used in calculating the future value of investments with consistent growth rates, relevant to concepts in Investment Strategies.

The Sum of an Infinite Geometric Series

If the absolute value of the common ratio (|r|) is less than 1 (i.e., -1 < r < 1), the geometric series converges to a finite sum as the number of terms approaches infinity. The formula for the sum (S_∞) of an infinite geometric series is:

S_∞ = a / (1 - r)

This formula is only valid when |r| < 1. If |r| ≥ 1, the series diverges and does not have a finite sum.

For example, consider the geometric series 1, 0.5, 0.25, 0.125, ... Here, a = 1 and r = 0.5.

S_∞ = 1 / (1 - 0.5) = 1 / 0.5 = 2

Therefore, the sum of this infinite geometric series is 2. This concept is vital in understanding the present value of a perpetuity, a concept in Valuation and finance.

Applications of Geometric Progressions

Geometric progressions appear in numerous real-world scenarios. Here are some examples:

**Compound Interest:** The growth of an investment with compound interest follows a geometric progression. The initial investment is 'a', the interest rate is used to calculate 'r', and the number of compounding periods determines 'n'. Understanding this is fundamental to Portfolio Management.
**Population Growth:** Under ideal conditions (unlimited resources), population growth can be modeled as a geometric progression.
**Radioactive Decay:** The decay of radioactive substances follows a geometric progression, with the initial amount as 'a' and the decay rate determining 'r'.
**Bouncing Ball:** The height a ball reaches after each bounce forms a geometric progression, with the initial height as 'a' and the coefficient of restitution determining 'r'.
**Fractals:** Many fractal patterns are based on geometric progressions.
**Financial Markets:** While markets aren't perfectly predictable, geometric progressions can help model certain trends and predict potential price movements, particularly when analyzing exponential growth or decay phases. This leads to applications in Trend Following strategies.
**Depreciation:** The value of an asset decreasing over time (e.g., a car) can be modeled using a geometric progression.
**Computer Science:** Binary representation of numbers is based on the geometric progression with a base of 2.