Class 10 Maths Arithmetic Progressions | Arithmetic Progression |

**Arithmetic Progression**

An arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, sequence 1, 3, 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2. Each of the numbers in the list is called a term.

**Real life applications**

- If you put 10 Rs every day on your piggy bank, you want to know the money after some x days
- If salary of a person is Rs 5000 & it increase by 1000 Rs every year, what will be salary after x years?
- The lengths of the rungs of a ladder decrease uniformly by 3 cm from bottom to top. If the bottom rung is 50 cm in length, how to find length of any rung?

An arithmetic progression is a list of numbers in which each term is obtained by adding a fixed number to the preceding term. E.g.: 2,4,6,8,10 …. This fixed number is called the common difference of the AP. This common difference can be positive, negative or zero

Positive common difference (3): 1,4,7,10,13..

Negative common difference (-1) : 7,6,5,4,3,2,1,0,-1 …

Zero Common Difference : 3,3,3,3,3,3,3,3,3,3….

Let us denote the first term of an AP by a_{1}, second term by a_{2}, . . ., nth term by a_{n} and the common difference by d. Then the AP becomes a_{1}, a_{2}, a_{3}, . . ., a_{n}.

So, a_{2} – a_{1} = a_{3} – a_{2} = . . . = a_{n} – a_{n }_{– 1} = d.

Thus a, a + d, a + 2d, a + 3d, . . . represents an arithmetic progression where a is the first term and d the common difference. This is called the general form of an AP.

AP with a finite number of terms are called a finite AP E.g.: {1,3,5,7}, while AP with infinite number of terms is called infinite AP, E.g.:{1,3,5,7,…}.

To know about an AP, the minimum information that we need is

- First term, denoted by ‘a’
- Common difference, denoted by ‘d’

With this we can form the AP as a, a + d, a + 2d, a + 3d, ……

For instance if the first term a is 5 and the common difference d is 2, then the AP is 5, 7,9, 11, . . .

Also, given a list, we can tell if it is AP or not.

List 2,4,6,8,10… here 4-2 = 6-4 = 8-6 = 10-8 =2, thus it is AP with common difference 2

List 1,3,4,6,7, here 3-1 ≠ 4-3, thus it is not AP.

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