### Theory:

Let us see a moment to look around our surrounding. 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.

In this chapter, we deep dive and explore the Arithmetic Progression with examples.

Let us see a scenario before we go into Arithmetic Progression.
Example:
Nancy is a teacher, got a job with a starting monthly pay of $$₹$$25000 and an annual raise of $$₹$$3000 per year. During the first, second, and third years, she will be paid $$₹$$25000, $$₹$$28000, and $$₹$$31000, respectively.

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

$$₹($$28000 $$–$$ 25000 $$–$$ $$) =$$ $$₹$$3000;

$$₹($$31000 $$–$$ 28000$$) =$$ $$₹$$3000.

Thus the difference between the successive numbers (salaries) is always $$₹$$3000.
From the above scenario, we can understand that the number sequence undergoes a certain sequence or a pattern by the constant common difference. And this kind of sequence is called 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:
Let us see few scenarios where we can observe the Arithmetic progression in real -life.

1. In a theatre, the arrangement of seats form 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 of the banks provide a fixed interest amount for saving account. For example, if we keep $$₹$$$$1000$$ in a saving account for one year, we would receive a specific percentage amount as interest at the end of the year. So, this saving amount plus interest amount added will become our new saving amount that will interest the proceeding years.

3. When you ride a taxi, you will be charged an initial rate and then a per mile or kilometres. And for every kilometre, you will be charged a certain amount extra plus the initial rate. This shows that 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 of Arithmetic Progression $$(A.P).$$ and practice the same concept.
Reference:
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