Sequence and Series

Sequence

A sequence is an ordered list of numbers, where each number is called a term. Sequences can follow specific rules or patterns.

Example: 

The sequence \( 2, 4, 6, 8, 10 \) follows the rule "add 2 to the previous term."

Types of Sequence based on the number of terms

1. Finite Sequence

A finite sequence has a limited number of terms. It begins and ends after a specific number of terms.

Example: 

The sequence \( 3, 6, 9, 12 \) is a finite sequence with 4 terms.

2. Infinite Sequence

An infinite sequence has an unlimited number of terms, and it continues indefinitely without stopping.

Example: 

The sequence \( 1, 2, 3, 4, 5, \dots \) continues forever and is an infinite sequence.

Series

A series is the sum of the terms of a sequence. When we add all the terms together, we create a series. It is represented by \(\sum + \text{ general term}\).

Example: 

If the sequence is \( 1, 2, 3 \), the series would be \( 1 + 2 + 3 = 6 \).

General Term (or nth Term)

The general term of a sequence, denoted as \( t_n \), represents a formula that gives the \( n \)-th term of the sequence. It helps to find any term in the sequence without listing all previous terms.

Example: 

For the sequence \( 2, 4, 6, 8, \dots \), the general term is \( t_n = 2n \), where \( n \) is the position of the term. So, the 5th term is \( t_5 = 2(5) = 10 \).

Types of Sequence based on the pattern (rule)

Arithmetic Sequence

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This difference is called the common difference.

Example: 

The sequence \( 5, 8, 11, 14, \dots \) has a common difference of \( 3 \), since each term is obtained by adding 3 to the previous one.

The \(n^{th}\) term of Arithmetic Sequence Formula:

If a = first term, n = \(n^{th}\) term and d = common difference, then \(n^{th}\) term is: 

Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence.

Example: 

If the sequence is \( 5, 8, 11 \), the arithmetic series would be \( 5 + 8 + 11 = 24 \).

Geometric Sequence

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

Example: 

The sequence \( 3, 6, 12, 24, \dots \) has a common ratio of \( 2 \), since each term is obtained by multiplying the previous term by 2.

The \(n^{th}\) term of geometric Sequence Formula:

If a = first term, n = \(n^{th}\) term and r = common ratio, then \(n^{th}\) term is: 

Geometric Series

A geometric series is the sum of the terms in a geometric sequence.

Example: 

If the sequence is \( 3, 6, 12 \), the geometric series would be \( 3 + 6 + 12 = 21 \).