## Introduction: Arithmetic Progressions

#### Sequence-

It is a set of numbers which follow a specific pattern and can be finite or infinite. For example, the sequence 1, 2, 3, 4, 5,… is an infinite sequence of natural numbers.

#### Series-

The sum of the numbers in a sequence is called a series. The series of natural numbers 1+2+3+4+5… provides an example.

#### Progression-

A progression is a sequence in which the general term can be can be expressed using a mathematical formula.

#### Arithmetic progression-

An arithmetic progression (AP) is a progression in which the difference** **between two consecutive terms is constant. Example: 2, 5, 8, 11, 14…. is an arithmetic progression.

#### The nth term of AP –

The nth term of an A.P is given by Tn= a+(n−1)d, where **a** is the first term, **d **is a common difference and **n **is the number of terms.

#### Sum of n terms of an AP –

Sn= n/2(2a+(n−1)d)

The sum of n terms of an A.P is also given by : **Sn= n/2(a+l)**

Where **a** is the first term, **l** is the last term of the A.P. and **n** is the number of terms.

#### Arithmetic mean (AM)-

It is the simple average of a given set of numbers. The AM is defined for any set of numbers. The numbers need not necessarily be in an A.P.

A.M= Sum of terms/Number of terms

#### Sum of first n natural numbers-

Sn=n(n+1)/2

This formula is derived by treating the sequence of natural numbers as an A.P where the first term (a) = 1 and the common difference (d) = 1.