Sequences
A set of numbers arranged in a definite order according to some rule is called a sequence.
Sequences following specific patterns are called progressions.
A sequence is called finite or infinite according as the number of terms in it is finite or infinite.
A finite sequence is described by $\displaystyle a_{1},a_{2},a_{3},a_{4},...a_{n}$ and an infinite sequence is described by $\displaystyle a_{1},a_{2},a_{3},a_{4},...\infty$.
Each number of the set is called a term. The $\displaystyle n^{th}$ term is the number at the $\displaystyle n^{th}$ position of the sequence and is also called the general term of the sequence.
Examples: Consider the sequence of numbers 2,4,8,16,32,64,.... Each term is obtained multiplying preceding term by 2. $\displaystyle n^{th}$ term of this sequence can be written as $\displaystyle a_{n}=2n$.
Consider the sequence 1,1,2,3,5,8,13,.... $\displaystyle a_{n}=a_{n-1}+a_{n-2} (n>2)$. This sequence is called Fibonacci sequence.
Series
If $\displaystyle a_{1},a_{2},a_{3},a_{4},...a_{n}...$ be a given sequence, then the expression $\displaystyle a_{1}+a_{2}+a_{3}+...+a_{n}+...$ is called the series associated with the given sequence.
The series is finite or infinite according as the given sequence is finite or infinite.
Series are often represented in compact form, called sigma notation, $\displaystyle \sum$
Thus the series $\displaystyle a_{1}+a_{2}+a_{3}+...+a_{n}$ is abbreviated as $\displaystyle \sum_{k=1}^{n}a_{k}$
A set of numbers arranged in a definite order according to some rule is called a sequence.
Sequences following specific patterns are called progressions.
A sequence is called finite or infinite according as the number of terms in it is finite or infinite.
A finite sequence is described by $\displaystyle a_{1},a_{2},a_{3},a_{4},...a_{n}$ and an infinite sequence is described by $\displaystyle a_{1},a_{2},a_{3},a_{4},...\infty$.
Each number of the set is called a term. The $\displaystyle n^{th}$ term is the number at the $\displaystyle n^{th}$ position of the sequence and is also called the general term of the sequence.
Examples: Consider the sequence of numbers 2,4,8,16,32,64,.... Each term is obtained multiplying preceding term by 2. $\displaystyle n^{th}$ term of this sequence can be written as $\displaystyle a_{n}=2n$.
Consider the sequence 1,1,2,3,5,8,13,.... $\displaystyle a_{n}=a_{n-1}+a_{n-2} (n>2)$. This sequence is called Fibonacci sequence.
Series
If $\displaystyle a_{1},a_{2},a_{3},a_{4},...a_{n}...$ be a given sequence, then the expression $\displaystyle a_{1}+a_{2}+a_{3}+...+a_{n}+...$ is called the series associated with the given sequence.
The series is finite or infinite according as the given sequence is finite or infinite.
Series are often represented in compact form, called sigma notation, $\displaystyle \sum$
Thus the series $\displaystyle a_{1}+a_{2}+a_{3}+...+a_{n}$ is abbreviated as $\displaystyle \sum_{k=1}^{n}a_{k}$
0 Comments