Arithmetic Progressions

5.1 Introduction

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

We will learn ourselves of arithmetic progressions and also learn about how to constructing an arithmetic progression (A.P). 

5.2 Arithmetic Progressions

Consider the following lists of numbers :

  1. 1, 2, 3, 4, . . .

  2. 100, 70, 40, 10, . . .

  3. −3, −2, −1, 0, . . .

  4. 3, 3, 3, 3, . . .

  5. −1.0, −1.5, −2.0, −2.5, . . .

Each of the numbers in the list is called a term.

An arithmetic progression (A.P.) is the list of numbers in which each term is obtained by adding or subtracting a fixed number to the preceding term except the first term.

This fixed number is called the common difference of the AP. Remember that it can be positive, negative or zero.

a, a + d, a + 2d, a + 3d, . . .

In above example represents an arithmetic progression where a is the first term and d the common difference. This is called the general form of an AP.

5.3 nth Term of an AP

We learn here the nth term a n of the AP with first term a and common difference d is given by an = a + (n − 1) d.

an is also called the general term of the AP. If there are m terms in the AP, then am represents the last term which is sometimes also denoted by l.

5.4 Sum of First n Terms of an AP

We will now use the same technique to find the sum of the first n terms of an AP :

a, a + d, a + 2d, . . .

The nth term of this AP is a + (n − 1) d. Let S denote the sum of the first n terms of the AP. We have

S = a + (a + d) + (a + 2d) + . . . + [ a + (n − 1) d ] (1)

Rewriting the terms in reverse order, we have

S = [ a + (n − 1) d ] + [ a + (n − 2) d ] + . . . + (a + d) + a (2)

On adding (1) and (2), term-wise. we get

2S = [ 2a + (n- 1)d ] + [ 2a + (n - 1)d ] + ... + [ 2a + (n - 1)d ] + [ 2a + (n - 1)d ]/n times

or, 2S = n [ 2a + (n − 1) d ] (Since, there are n terms)

or, S =2/n [ 2a + (n − 1) d ]

So, the sum of the first n terms of an AP is given by

S = n/2 [ 2a + (n − 1) d ]

S = n/2(a + an )

If there are only n terms in an AP, then an = l, the last term.

S = n/2(a + l )

The sum of first n positive integers is given by

Sn = n(n + 1)/2