## What is an Arithmetic Progression (AP)?

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant difference is known as the “common difference.” The general form of an arithmetic progression can be written as:

where:

- is the first term.
- is the common difference.

For example, in the sequence , the common difference is , and the first term is .

## Finding the Common Difference

To identify whether a sequence is an arithmetic progression and to find the common difference, you can subtract any term from the term that follows it. Mathematically, this can be written as:

For example, consider the sequence :

- Common difference
- Common difference
- Common difference

In this case, the sequence is indeed an arithmetic progression with a common difference of .

## General Term of an AP

The general term (or the -th term) of an arithmetic progression can be found using the formula:

where:

- is the -th term.
- is the first term.
- is the common difference.
- is the position of the term in the sequence.

For instance, finding the 5th term of the arithmetic sequence involves:

Therefore, the 5th term is .

## Sum of the First Terms of an AP

The sum of the first terms of an arithmetic progression can be found using the formula:

Alternatively, this can also be written as:

where:

- is the sum of the first terms.
- is the first term.
- is the common difference.
- is the -th term.

For example, to find the sum of the first 5 terms of the sequence :

Therefore, the sum of the first 5 terms is .

## Real-World Examples of Arithmetic Progressions

Arithmetic progressions can be found everywhere in daily life. Here are a few examples:

**Bank Savings:**If you save a fixed amount of money every week or month, the total amount of money saved over time forms an arithmetic progression.**Floor Tiles:**If floor tiles are laid in a pattern where each row has one more tile than the previous, they form an arithmetic progression.**Staircases:**The number of steps taken to reach each successive floor in a multi-storey building can form an arithmetic sequence if each floor has the same number of steps.

Understanding arithmetic progressions allows us to analyze and solve many problems in our daily lives.

## Conclusion

Arithmetic progressions are a fundamental concept in mathematics, relevant both in academics and in various real-world applications. By understanding the properties and formulas related to arithmetic progressions, students can enhance their problem-solving skills and appreciate the omnipresence of mathematics in everyday life.

Key points to remember include:

- The common difference ()
- The formula for the -th term:
- The formula for the sum of the first terms:

With this solid foundation, students are well-prepared to explore more complex mathematical concepts and their applications.