The series or sequence which are obtained by adding or subtracting number to the previous term is called arithmetical progression. It is also written as AP. The finite sequence a, b, c, d, … or infinite sequence a, b, c, d, … will be arithmetical progression if the differences b – a, c – b, d – c, … always remain constant. Normally the sequence a, a + d, a + 2d, a + 3d, … + a + nd, a + (n + 1)d is an arithmetic progression with first term a and the common difference d. Generally, a and d are the notation for first term and common difference of an AP. The last term of a finite AP is denoted by l and it is represented by l = a + (n – 1)d, where n is the number of terms of AP. The difference between the term and the term previous to it called common difference of AP. The common difference is constant for a arithmetical progression. An arithmetic progression having a common difference of zero is called a constant arithmetic progression.

## What is a Progression

When a sequence or series be written in by some definite rule we get a progression. We must remember that each progression is a series but each series is need not be a progression.

### What is a sequence

A sequence is a function in a non-empty set whose domain and range are the natural numbers. Any set of numbers, which are written in a definite order and made of some definite rule, is called a sequence of numbers.

#### What is series

When the elements of a sequence be written by putting positive or negative sign in between the numbers, then we get a series of numbers. For example, 2 + 4 + 6 + 8 + 10 + … is a series. If the number of terms in a series are finite then the series is called a finite series but if the number of terms are infinite, the series is called an infinite series.

##### Arithmetic Mean

If three numbers be in AP then the middle number is called arithmetic mean. It is denoted by A.M. If a, A, b are in A.P. then A – a = b – A or A = ½ (a + b). Here, A is the arithmetic mean between a and b.

#### History of Arithmetic Progression

Evidence is found that Babylonians some 400 years ago, knew of Arithmetic and geometric progressions in Mathematics. According to Boethins (570 AD), these progressions were known to early Greek writers. Among the Indian mathematicians, Aryabhata (470 AD) was the first to give formula for the sum of squares and cubes of natural number in his famous work Aryabhatiyam written around 499 A.D. He also gave the formula for finding the sum of n terms of an Arithmetic Progression starting with pth term. Indian mathematician Brahmagupta (598 AD), Mahavira (850 AD) and Bhaskara (1114-1185 AD) also considered the sums of squares and cubes.

An Italian Mathematician Leonardo Pisano Bogollo (also known as Leanardo of Pisano, Leonardo Bonacci, Leonardo Finonacci) is popular for a sequence of number called FIBONACCI SEQUENCE, 0, 1, 1, 2, 3, 5, 6, … where every term of the sequence (except first and second) is the sum of two numbers before it. He also helped to spread the use of Hindu Arabic numerals to replace Roman numerals.

##### Some common sums based on Arithmetic Progression

- Usha applied for a job and got selected. She has been offered a job with a starting monthly salary of ₹8000, with an annual increment of ₹500. Her salary (in rupees) for to 1st, 2nd, 3rd … years will be 8000, 8500, 9000 …. respectively.
- The lengths of the rungs of a ladder decrease uniformly by 2 cm from bottom to top. The bottom rung is 45 cm in length. The lengths (in cm) of the 1st, 2nd, 3rd, …. 8th rung from the bottom to the top are 45, 43, 41, 39, 37, 35, 33, 31 respectively.
- In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?
- In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?
- A sum of ₹700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of the prizes.

### What is an arithmetic progression?

The series or sequence which are obtained by adding or subtracting number to the previous term is called arithmetical progression. For example, 1 + 3 + 5 + 7 + ….

### Who is known as the father of arithmetic progression?

The concepts of Arithmetic Progression were invented by Johann Carl Friedrich Gauss at his school level when he was trying to find out the sum of first 100 natural number.

### What is the formula for the sum of n terms of an AP?

The sum of an arithmetic progression can be determined using the following formulae:

S = n/2[a + l], where n is the number of terms, a is the first term and l is the last term.

S = n/2[2a + (n – 1) d], where d is the common difference.

### What is the general formula for an Arithmetic Progression?

The general formula for an AP is Tn = a + (n – 1) d, where a is the first term, n is number of terms and d is the common difference.