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

1. If you put 10 Rs every day on your piggy bank, you want to know the money after some x days
2. If salary of a person is Rs 5000 & it increase by 1000 Rs every year, what will be salary after x years?
3. 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 a₁, second term by a₂, . . ., nth term by a_n and the common difference by d. Then the AP becomes a₁, a₂, a₃, . . ., a_n.

So, a₂ – a₁ = a₃ – a₂ = . . . = 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

1. First term, denoted by ‘a’
2. 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.