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.