Sequences

A sequence $a$ is a list of numbers $a_1, a_2, ...$ . The notation is:

(a_n) = (a_1, a_2, a_3, ..., a_n, a_{n+1}, ... )

The numbers $a_n$ $(n=1,2,...)$ are called terms, where $a_1$ is the first term, $a_2$ the second term, and $a_n$ the $n$ -th term. A formula that describes how to directly calculate the $n$ -th term is called an explicit law. A formula that describes how to get from one element to the next in the sequence is called a recursive law.

The brackets indicate that sequences are tuples, like coordinates $(x\vert y\vert z)$ , so the order of the numbers is essential: $(1|2|3)\neq (2|1|3)$ . In contrast, numbers in curly brackets are sets, where the order is not important. For example, $\{1,2,3\}=\{2,1,3\}$ .
The indices of $a$ denote the position of the term in the sequence. $a_1$ is the first fist term, $a_9$ is the 9-th term. $a_n$ is the $n$ -th term and $a_{n+1}$ is the term that follows after the $n$ -th term.

Example 1

Find the explicit and recursive laws for the sequences:
1. $(a_n)=(1,3,5,...)$
2. $(b_n)=(1, 0.5, 0.25, 0.125, ...)$
3. $(c_n)=(1,-1,1,-1,...)$
4. $(d_n)=(1,2,4,8,...)$
Determine the first 3 terms of the sequence
1. $a_n=3n$ ( $n=1,2,...$ )
2. $b_1=2, b_{n+1}=2b_n+1$ ( $n=1,2,3...$ )

Solution

It is
1. explicit: $a_n=2n-1\,\,(n=1,2,3,...)$ , recursive: $a_1=1$ , $\underbrace{a_{n+1}}_{new}=\underbrace{a_n}_{old}+2 \,\,(n=1,2,3,...)$
2. explicit: $b_n=\left(\frac{1}{2}\right)^{n-1}\,\,(n=1,2,3,...)$ , recursive: $b_1=1$ , $\underbrace{b_{n+1}}_{new}=\frac{1}{2}\underbrace{b_n}_{old} \,\,(n=1,2,3,...)$
3. explicit: $c_n=(-1)^{n-1}\,\,(n=1,2,3,...)$ , recursive: $c_1=1$ , $\underbrace{c_{n+1}}_{new}=-1\cdot \underbrace{c_n}_{old} \,\,(n=1,2,3,...)$
4. explicit: $d_n=2^{n-1}\,\,(n=1,2,3,...)$ , recursive: $d_1=1$ , $\underbrace{d_{n+1}}_{new}=2\cdot \underbrace{d_n}_{old} \,\,(n=1,2,3,...)$
It is
1. $a_1=3, a_2=6, a_3=9$
2. $b_1=2, b_2=2\cdot 2+1=5, b_3=2\cdot 5+1 =11$