Arithmetic Progressions

An AP is a pattern

An AP is a pattern in which each term except the first is obtained by adding a fixed number to the previous term. The standard form is a, a + d, a + 2

The first term is denoted by

The first term is denoted by a. The common difference is denoted by d and is found by subtracting consecutive terms: d = t_2 - t_1 = t_3 - t_2 = ...

The nth term is given by

The nth term is given by t_n = a + (n - 1)d. This formula gives any term directly when the term number n, first term a, and common difference d are kn

The sum of the first n

The sum of the first n terms is S_n = n/2[2a + (n - 1)d]. It can also be written as S_n = n/2(a + l) when the last term l is known.

The sum of the first 100

The sum of the first 100 natural numbers is found by writing the series forwards and backwards and adding term by term. This method leads to the AP su