**Sequence and Series:** One of the most significant concepts in Arithmetic is Sequence and Series. A sequence represents a collection of elements that can be repeated in any order. In simpler terms, a sequence is a systematic arrangement of objects or items in a sequential manner.

For instance: a1, a2, a3, a4, …

On the other hand, a series refers to the sum of all the elements within a sequence. In essence, a series is the cumulative total of all the elements present in a sequence. An example commonly used to explain sequence and series is Arithmetic Progression (AP).

For example: If a1, a2, a3, a4, … is considered a sequence, then the sum of terms in the sequence a1 + a2 + a3 + a4, … is considered a series.

**Concept for Sequences:**

**Sequence:**An ordered list of numbers following a certain pattern.**Term:**Each individual number in a sequence.**Arithmetic Sequence:**A sequence where the difference between consecutive terms is constant.**Geometric Sequence:**A sequence where the ratio between consecutive terms is constant.**Fibonacci Sequence:**A famous sequence where each term is the sum of the two preceding terms, starting with 0 and 1.**Recursive Formula:**A formula that expresses a term in a sequence in terms of previous terms.**Explicit Formula:**A formula that directly gives the value of the nth term in a sequence without relying on previous terms.

**Concept** for Series:

**Concept**for Series:

**Series:**The sum of the terms in a sequence.**Arithmetic Series:**The sum of an arithmetic sequence.**Geometric Series:**The sum of a geometric sequence.**Convergence:**A series is said to converge if the sum of its terms approaches a finite value as more terms are added.**Divergence:**A series is said to diverge if the sum of its terms does not approach a finite value.**Partial Sum:**The sum of a finite number of terms in a series.**Infinite Series:**A series that has an infinite number of terms.

**Important Concepts:**

**Sum of an Arithmetic Series:**S_n = (n/2) * (a + l), where S_n is the sum of the first n terms, a is the first term, and l is the last term.**Sum of a Geometric Series:**S_n = a * (1 – r^n) / (1 – r), where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.**Telescoping Series:**A series where most terms cancel out, leaving only a few terms to compute the sum.**Alternating Series Test:**A test used to determine whether an alternating series converges.**Ratio Test:**A test for the convergence of a series based on the ratio of consecutive terms.**Integral Test:**A test for the convergence of a series by comparing it to an integral.

These are just some of the keywords and concepts related to sequences and series in mathematics. They play a crucial role in various fields, including calculus, number theory, and mathematical analysis.

### Types of Sequence and Series

Some of the most common examples of sequences are:

- Arithmetic Sequences
- Geometric Sequences
- Harmonic Sequences
- Fibonacci Numbers

**Arithmetic Sequences**

A sequence where each term is generated by adding or subtracting a fixed number to/from the previous term.

**Geometric Sequences**

A sequence where each term is obtained by multiplying or dividing the previous term by a constant value.

**Harmonic Sequences**

A series of numbers forms a harmonic sequence if the reciprocals of all its elements form an arithmetic sequence.

**Fibonacci Numbers**

Fibonacci numbers constitute an intriguing sequence in which each element is the sum of the two preceding elements, starting with 0 and 1. The sequence is defined as F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-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=a+b2**

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}**= 1a+(n–1)d**

The harmonic mean between a and b is **H.M=2aba+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} = n2(a+l),

S_{n} = n2[2a+(n–1)d]

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

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

The sum S of infinite geometric series is **S=a1–r;**

Difference Between Sequences and Series

Let us find out how a sequence can be differentiated with series.

Sequences | Series |

Set of elements that follow a pattern | Sum of elements of the sequence |

Order of elements is important | Order of elements is not so important |

Finite sequence: 1,2,3,4,5 | Finite series: 1+2+3+4+5 |

Infinite sequence: 1,2,3,4,…… | Infinite Series: 1+2+3+4+…… |

**Example**

**Write an A.P when its first term is 10 and the common difference is 3.**

**Solution:**

**Step 1: **Arithmetic Progression = A.P. = a, a+m , a+2m , a +3m, a+4m…….

**Step 2:** here, a=10 and m = 3

So let put its value in the equation

**Step 3: **10, 10+3, 10 +2*3, 10 + 3*3, 10 + 4 * 3 ………

We get,

10,13,16, 19, 22, …….

**Example 2**

**Write the first three terms of the sequence defined as a**_{n }**= n**^{2 }**+ 1**

**Solution:**

**Step1:** we have a_{n }= n^{2 }+ 1

**Step 2: **Putting n = 1,2,3. We get

**Step 3:** a_{1 }= 1^{2 }+ 1 = 1 + 1 = 2

a_{2} = 2^{2 }+ 1 = 4 + 1 = 5

a_{3 }= 3^{2 }+ 1 = 9 + 1 = 10

**Quiz Time**

- Write the first five terms of each of the following sequence whose nth terms are:

- a
_{n}= 3n + 2 - a
_{n}= 3 n

- Write an A.P. having 4 as the first term and -3 as the common difference.
- Write a G.P. having 4 as the first term and 2 as the common ratio.

**Summary Points**

- In a math series and sequence, the first term is denoted as “a,” the common difference is represented as “d,” and the nth term is identified as “an.”
- The arithmetic sequence can be explained as a, a + d, a + 2d, a + 3d, …
- Obtaining each term in a geometric progression involves multiplying the preceding term by the common ratio of the successive term.
- The geometric progression formula is known as :

a_{n} = ar^{n-1}

- The sum of infinite geometric progression formula is known as :

Sn = a/(1-r) Where |r| < 1. Read Also: Cube and Dice