Chapter 5

Introduction to Arithmetic Progression

Sequence: A sequence is an ordered list of objects.

Progressions: Sequences, which follows specific patterns, are called progressions.

Arithmetic progression (A.P.): An arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

The constant is called the common difference of the arithmetic progression and it can be negative, positive or zero.

General form of an A.P.: 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.

In an A.P., every succeeding term is obtained by adding common difference to the preceding term. So, common difference found by subtracting any term from its succeeding term.

In general, for an A.P. a₁, a₂… a_n, we have d = a_n₊₁ – a_n

Where a_n₊₁ and a_n are the (n + 1)^th and the n^th terms respectively.

To find the common difference, we should subtract the n^th term from the (n + 1)^th term even if the (n + 1)^th term is smaller.

Nth term of an A.P.: The n^th term a_n of the A.P. with first term a and common difference d is given by a_n = a + (n – 1) d.

Sum of first n terms of an A.P.: The sum of the first n terms of an A.P. is given by

S = [2a + (n – 1) d]

The sum of first n positive integers is given by

S_n =