## Understanding Sequences

Before we dive into geometric progressions, let’s understand what sequences are. A sequence is a list of numbers in a specific order. Each number in a sequence is called a term. Sequences can be finite or infinite. For example, the sequence 2, 4, 6, 8 is finite, whereas the sequence 1, 2, 3, 4, 5, … is infinite.

There are various types of sequences in mathematics, each following a specific rule. One common type is the arithmetic sequence, where the difference between consecutive terms is constant. However, this article focuses on geometric sequences.

## Defining a Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as *r*. A geometric progression can be represented as:

Here, *a* is the first term, and *r* is the common ratio.

For example, consider the sequence 3, 6, 12, 24, 48. Each term is obtained by multiplying the previous term by 2. So, this is a geometric progression with the first term and the common ratio .

## Finding the n-th Term of a Geometric Progression

To find the n-th term of a geometric progression, we use the formula:

Where

- is the n-th term,
- is the first term,
- is the common ratio, and
- is the term number.

For example, in the geometric sequence 3, 6, 12, 24, 48, to find the 5th term:

Thus, the 5th term is 48.

## Sum of the First n Terms

The sum of the first n terms of a geometric progression (denoted as ) is given by the formula:

If ,

Where:

- is the sum of the first n terms,
- is the first term,
- is the common ratio, and
- is the number of terms.

For example, to find the sum of the first 5 terms of the sequence 3, 6, 12, 24, 48:

Thus, the sum of the first 5 terms is 93.

## Real-World Applications of Geometric Progressions

Geometric progressions are not just theoretical concepts; they have practical applications in various real-world situations. Here are a few examples:

### Population Growth

In biology, geometric progressions can model population growth. If a population grows at a fixed percentage rate per year, then it follows a geometric progression.

### Finance

In finance, compound interest is calculated using geometric progressions. The amount of money in an account where interest is compounded annually follows a geometric sequence.

### Physics

In physics, certain phenomena such as radioactive decay can be modeled using geometric progressions. The quantity of a radioactive substance decreases geometrically over time.

## Conclusion

Geometric progressions are a fundamental concept in mathematics, with various applications in real life. By understanding the properties and formulas associated with geometric progressions, students can solve complex problems and appreciate the patterns in the world around them. Whether it’s calculating population growth or understanding financial interest, geometric progressions provide a powerful tool for analysis and prediction.

As we continue to explore the fascinating world of mathematics, the study of sequences and series remains a cornerstone, opening up countless possibilities for both theoretical exploration and practical application.