# Sequence and Series Important Formulas for Competitive Exams

### Arithmetic Progression(AP)

Arithmetic progression(AP) or arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, d to the preceding term. The constant d is called common difference.

An arithmetic progression is given by a, (a + d), (a + 2d), (a + 3d), …
where a = the first term , d = the common difference

Examples

1, 3, 5, 7, … is an arithmetic progression (AP) with a = 1 and d = 2

7, 13, 19, 25, … is an arithmetic progression (AP) with a = 7 and d= 6

 If a, b, c are in AP, 2b = a + c

### nth term of an arithmetic progression

 tn = a + (n – 1)d where tn = nth term, a= the first term , d= common difference

Example 1
Find 10th term in the series 1, 3, 5, 7, …

a = 1
d = 3 – 1 = 2

10th term, t10 = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19

Example 2
Find 16th term in the series 7, 13, 19, 25, …

a = 7
d = 13 – 7 = 6

16th term, t16 = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97