TL;DR — Quick Summary
Video Lessons
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
The general term (or nth 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 \( 3 + 6 + 12 = 21 \).