## Introduction to Geometric Progressions

A **geometric progression** (or geometric sequence) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the *common ratio*. If the first term of the sequence is denoted as and the common ratio is , the sequence looks like this:

Geometric progressions are a fundamental concept in mathematics due to their simplicity and the interesting properties they exhibit, making them highly applicable to various real-life scenarios.

## Finance and Economics

One of the most notable applications of geometric progressions is in finance, particularly in the calculation of compound interest. When money is invested or borrowed at compound interest, the amount of money grows in a geometric progression.

Compound interest can be represented by the formula:

where:

- is the amount of money accumulated after years, including interest.
- is the principal amount (the initial amount of money).
- is the annual interest rate (decimal).
- is the number of times that interest is compounded per unit year.
- is the time the money is invested or borrowed for, in years.

For example, consider an investment of $1000 with an annual interest rate of 5% compounded annually. Using the compound interest formula, the value of the investment after 3 years can be calculated as follows:

## Population Growth

Population growth can often be modelled using geometric progressions, particularly in cases where the growth rate is constant. The population at a given time can be determined using the formula:

where:

- is the population at time .
- is the initial population.
- is the growth rate per time period.
- is the number of time periods.

For instance, if a town has an initial population of 10,000 people and the population grows by 2% per year, the population after 5 years would be:

Thus, the population would be approximately 11,041 after 5 years, assuming a constant growth rate.

## Physics and Engineering

Geometric progressions are also prevalent in the fields of physics and engineering. For example, they are used to calculate the distances between specific positions in uniformly accelerating motion or to determine the resistance in a circuit with resistors arranged in a geometric sequence.

Consider an example of a ball bouncing to a fraction of its previous height. If a ball is dropped from a height and bounces to a height of each time, the height of the ball after each bounce follows a geometric progression:

This sequence has a common ratio of .

## Computer Science

In computer science, geometric progressions are utilized in algorithms and data structures. An example is the exponential growth of time complexity in algorithms, such as in the case of recursive algorithms solving problems of size by breaking them into two subproblems of size .

Consider the *binary search algorithm*, where the search interval is halved in each step. The number of steps required to search for an element in a sorted array of elements can be represented as:

which results in a logarithmic time complexity , showing geometric progression properties.

## Environmental Science

Geometric progressions also appear in environmental science, particularly in the modeling of resource depletion or growth of algae in water bodies. If a population of algae doubles every week, and the initial population is 1000, the population of algae can be described by a geometric progression:

where is the population after weeks.

Therefore, after 4 weeks, the population would be:

Thus, the population would exponentially grow to 16,000 in just 4 weeks.