## Introduction to Geometric Progression

A Geometric Progression (GP) 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. For example, in the sequence 2, 6, 18, 54, …, each term is obtained by multiplying the previous term by 3.

Mathematically, a geometric progression can be written as:

where:

**a**is the first term,**r**is the common ratio,**ar^n**is the nth term.

## Formula for the Sum of First n Terms

The sum of the first n terms of a geometric progression can be calculated using the formula:

where:

**S_n**is the sum of the first n terms,**a**is the first term,**r**is the common ratio,**n**is the number of terms.

If the common ratio , the sum of the first n terms is simply:

since every term in the progression is the same.

## Derivation of the Formula

Let’s derive the formula for the sum of the first n terms of a geometric progression. Consider the sum:

Multiplying both sides of the equation by the common ratio , we get:

Now, we subtract the second equation from the first equation:

Factoring out on the left-hand side:

Finally, solving for , we get:

## Example: Saving Money

Imagine you decide to save money starting with $10 in the first month, and each month you save double the amount you saved the previous month. This forms a geometric progression where:

- The first term
- The common ratio

For example:

To find the total amount saved after 6 months, we use the formula for the sum:

So, after 6 months, you will have saved $630.

## Further Applications

Geometric progressions are not just limited to financial calculations. They appear in various fields such as physics, biology, and computer science. For instance, in population biology, if a population grows by a fixed percentage each year, the population size over multiple years forms a geometric progression.

As another example, consider a ball that bounces back to a fraction of its previous height after each bounce. If we want to find the total distance the ball travels, we can model the distances using a geometric progression.

## Conclusion

Understanding the sum of the first n terms of a geometric progression is essential in various real-world applications, from financial planning to analyzing scientific data. The formula provides a quick and effective way to calculate these sums, especially when dealing with a large number of terms.

The ability to recognize and apply geometric progressions enhances problem-solving skills and mathematical reasoning, which are valuable assets in both academic and everyday scenarios.