A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. A series is the addition of all the terms of a sequence. Sequence and series are similar to sets but the difference between them is in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms, which can be either finite or infinite. Let us start learning Sequence and series formula.

**Sequence and Series Formula**

**What are Sequence and Series?**

A sequence is an ordered list of numbers. The numbers in the list are the terms of the sequence. The terms of a sequence usually name as *a _{i }*or

*a*, with the subscripted letter

_{n}*i*or

*n*being the index. So, the second term of a sequence might be named

*a*

_{2}, and

*a*

_{12}would be the twelfth term.

A series termed as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.

S_{N} = a_{1}+a_{2}+a_{3} + .. + a_{n}

### Types of Sequence and Series

**Sequences:** A finite sequence stops at the end of the list of numbers like a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}……a_{n. }whereas, an infinite sequence is never-ending i.e. a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}……a_{n….}.

**Series: **In a finite series, a finite number of terms are written like a_{1 }+ a_{2 }+ a_{3} + a_{4 }+ a_{5} + a_{6 + }……a_{n}. In case of an infinite series, the number of elements are not finite i.e. a_{1 }+ a_{2 }+ a_{3} + a_{4 }+ a_{5} + a_{6 + }……a_{n }+_{…..}

### Some Common Sequences

**Arithmetic Sequences:**

A sequence in which every term is obtained by adding or subtraction a definite number to the preceding number is an arithmetic sequence.

**Geometric Sequences:**

A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

**Harmonic Sequences:**

If the reciprocals of all the elements of the sequence form an arithmetic sequence then the series of numbers is said to be in a harmonic sequence.

**Fibonacci Numbers:**

Fibonacci numbers form a sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F_{0} = 0 and F_{1} = 1 and F_{n} = F_{n-1} + F_{n-2}

**Sequence and Series Formulas**

The sequence of A.P: The n^{th} term a_{n} of the Arithmetic Progression (A.P) a, a+d, a+2d,…a, a+d, a+2d,… is given by

a_{n}=a+(n–1)d

Where,

a | First-term |

d | Common difference |

n | Position of the term |

l | Last term |

Arithmetic Mean: The arithmetic mean between a and b is given by **A.M=\(\frac{a+b}{2}\)**

The sequence of G.P: The nth term a_{n} of the geometric progression a, ar, ar^{2}, ar^{3},…, is **a _{n}=ar^{n}–1an=ar^{n–1}**

The geometric mean between a and b is **G.M= ±\sqrt{ab} **

Sequence of H.P: The n^{th} term a_{n} of the harmonic progression is **a _{n}= \( \frac{1}{a+(n–1)d} \)**

The harmonic mean between a and b is **H.M=\(\frac{2ab}{a+b}\)**

Series of A.P: If S_{n} denotes the sum up to n terms of A.P. a, a+d, a+2d,…a, a+d, a+2d,… then

S_{n} = \(\frac{n}{2}(a+l),\)

S_{n} = \( \frac{n}{2} [2a+(n–1)d] \)

The sum of n A.M between a and b is **A.M = \(\frac{n(a+b)}{2}\)**

Series of G.P: If S_{n} denotes the sum up to n terms of G.P is **S _{n}=\(\frac{a(1–rn)}{1–r}\); r≠1 and l=ar^{n}**

The sum S of infinite geometric series is **S=\(\frac{a}{1–r}; \)**

**Solved Examples**

#### Question 1: If 1, 3, 5, 7, 9…… is a sequence, Find Common difference, nth term, 21st term

Solution: Given sequence is, 1, 3, 5, 7, 9……

- a) common difference d = 3 – 1 = 2
- b) The nth term of the arithmetic sequence is denoted by the term T
_{n}and is given by T_{n}= a + (n-1) d, - c) 21
^{st}term as: T_{21}= 1 + (21-1)2 = 1+40 = 41.