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Let us take a moment to look around our surroundings. We might have observed specific patterns in nature, such as the patterns in the sunflowers, seashell covers, and a honeycomb's holes. (Click

*to know more*)We can observe similar patterns in mathematics as well. For instance, say the Fibonacci series, the Arithmetic Progression, and the Geometric Progression.

This chapter delves into and illustrates the arithmetic progression.

Let us see a scenario before we go into Arithmetic Progression.

Example:

Nancy, a teacher, got a job with a starting monthly pay of \(₹\)20000 and an annual raise of \(₹\)5000 per year. During the first, second, and third years, she will be paid \(₹\)20000, \(₹\)25000, and \(₹\)30000, respectively.

Now we calculate the difference of the salaries for the successive years, we get:

\(₹(\)25000 \(–\) 20000 \(–\) \() =\) \(₹\)5000;

\(₹(\)30000 \(–\) 25000\() =\) \(₹\)5000.

Thus the difference between the successive numbers (salaries) is always \(₹\)5000.

From the above scenario, we can understand that the number sequence follows a certain pattern due to the constant common difference. This type of sequence is known as arithmetic progression.

Arithmetic Progression:

The sequence of numbers in which each term is differ by the common difference from the previous term through out the sequence is known as arithmetic progression.

- Let \(a\) and \(d\) be real numbers. Then the numbers of the form $a,a+d,a+2d,a+3d,a+4d,a+5d$,... is said to form Arithmetic progression. And it is denoted by \(A\).\(P\).
- The number ‘\(a\)’ is called the first term, and ‘\(d\)’ is called the common difference.

Arithmetic Progression in real-life:

**1**. In a theatre, the arrangement of seats forms an arithmetic progression. For instance, the first row might have \(12\) seats; the second row has \(14\) seats, and the third row contains \(16\).

**2**. Most banks provide a fixed interest amount for savings accounts. For example, if we keep \(₹\)\(1000\) in a savings account for one year, we would receive a specific percentage amount as interest at the end of the year. So, this saving amount plus the interest amount added will become our new saving amount that will grow over the proceeding years.

**3**. When you take a taxi, you will be charged an upfront fee followed by a per mile or per kilometre fee. And for every kilometre, you will be charged a certain amount extra plus the initial rate. This shows how the arithmetic sequence works when determining the cost.

**4**. The lowest temperatures (in degrees Celsius) recorded in a city for a week in December, given in ascending order are:

\(– 4.1, – 3.8, – 3.5, – 3.2, – 2.9, – 2.6, – 2.3\)

**Now we learned about the fundamentals of Arithmetic Progression**\((A.P)\).

**In the upcoming lessons, we will learn the terms for Arithmetic Progression**\((A.P).\)

**and practice the same concept.**

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