Types of sequence

Arithmetic Sequences

Definition:

An arithmetic sequence is a sequence in which each term is obtained by adding a constant difference $d$ to the previous term. The general form of an arithmetic sequence is:

a_{n} = a_{1} + (n - 1) d

where:

$a_{n}$ is the $n$ -th term.
$a_{1}$ is the first term.
$d$ is the common difference between consecutive terms.

Example:

Consider the arithmetic sequence with $a_{1} = 3$ and $d = 2$ . The sequence is:

3, 5, 7, 9, 11, \dots

Using the formula for the $n$ -th term:

a_{n} = 3 + (n - 1) \cdot 2 = 2 n + 1

Sum of an Arithmetic Sequence:

The sum of the first $n$ terms of an arithmetic sequence (arithmetic series) is given by:

S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})

or equivalently:

S_{n} = \frac{n}{2} \cdot (2 a_{1} + (n - 1) d)

Geometric Sequences

Definition:

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant ratio $r$ . The general form of a geometric sequence is:

a_{n} = a_{1} \cdot r^{n - 1}

where:

$a_{n}$ is the $n$ -th term.
$a_{1}$ is the first term.
$r$ is the common ratio between consecutive terms.

Example:

Consider the geometric sequence with $a_{1} = 2$ and $r = 3$ . The sequence is:

2, 6, 18, 54, 162, \dots

Using the formula for the $n$ -th term:

a_{n} = 2 \cdot 3^{n - 1}

Sum of a Geometric Sequence:

The sum of the first $n$ terms of a geometric sequence (geometric series) is given by:

S_{n} = a_{1} \cdot \frac{1 - r^{n}}{1 - r}, r \neq 1

The sum of an infinite geometric series is:

S = \frac{a_{1}}{1 - r}, | r | < 1

Harmonic Sequences

Definition:

A harmonic sequence is a sequence where each term is the reciprocal of the corresponding term in an arithmetic sequence. The general form of a harmonic sequence is:

a_{n} = \frac{1}{n}

or, more generally:

a_{n} = \frac{1}{a_{1} + (n - 1) d}

where $a_{1}$ is the first term and $d$ is the common difference of the corresponding arithmetic sequence.

Example:

The harmonic sequence corresponding to the arithmetic sequence $1, 2, 3, 4, \dots$ is:

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots

Harmonic Series:

The sum of the harmonic sequence (the harmonic series) is:

\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots

This series diverges, meaning it grows without bound as more terms are added, even though each term gets smaller.

Recursive Sequences

Definition:

A recursive sequence is defined by a recurrence relation, where each term is a function of one or more of the previous terms. Instead of providing an explicit formula for $a_{n}$ , a recursive sequence gives a relationship between $a_{n}$ and previous terms like $a_{n - 1}$ , $a_{n - 2}$ , etc.

Example:

The sequence defined by the recurrence relation:

a_{n} = a_{n - 1} + 2, a_{1} = 3

is a recursive arithmetic sequence where each term is the previous term plus 2. The sequence is:

3, 5, 7, 9, 11, \dots

General Recursive Formula:

Many recursive sequences are defined in terms of both initial conditions and a recursive formula.

Example (Geometric Recursive):

a_{n} = r \cdot a_{n - 1}, a_{1} = 2

With $r = 3$ , the sequence is:

2, 6, 18, 54, \dots

Fibonacci Sequence

Definition:

The Fibonacci sequence is a famous recursive sequence where each term is the sum of the two preceding terms. The sequence is defined by the recurrence relation:

F_{n} = F_{n - 1} + F_{n - 2}, F_{1} = 1, F_{2} = 1

The first few terms of the Fibonacci sequence are:

1, 1, 2, 3, 5, 8, 13, 21, 34, \dots

Properties of the Fibonacci Sequence:

The Fibonacci sequence grows exponentially as $n$ increases.
The ratio of consecutive Fibonacci numbers approaches the golden ratio:

lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}} = \frac{1 + \sqrt{5}}{2} \approx 1.618