Definition:Progression

Definition

The word progression is a term used for either sequence or series in the following specific contexts.

Because its usage is in general ambiguous, it is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Arithmetic Progression

The term arithmetic progression is used to mean one of the following:

Arithmetic Sequence

An arithmetic sequence is a finite sequence $\sequence {a_k}$ in $\R$ or $\C$ defined as:

$a_k = a_0 + k d$ for $k = 0, 1, 2, \ldots, n - 1$

Thus its general form is:

$a_0, a_0 + d, a_0 + 2 d, a_0 + 3 d, \ldots, a_0 + \paren {n - 1} d$

Arithmetic Series

An arithmetic series is a series whose underlying sequence is an arithmetic sequence:

\(\ds S_n\)

\(=\)

\(\ds \sum_{k \mathop = 0}^{n - 1} a + k d\)

\(\ds \)

\(=\)

\(\ds a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}\)

Geometric Progression

The term geometric progression is used to mean one of the following:

Geometric Sequence

A geometric sequence is a sequence $\sequence {x_n}$ in $\R$ defined as:

$x_n = a r^n$ for $n = 0, 1, 2, 3, \ldots$

Thus its general form is:

$a, ar, ar^2, ar^3, \ldots$

and the general term can be defined recursively as:

$x_n = \begin{cases}

a & : n = 0 \\ r x_{n-1} & : n > 0 \\ \end{cases}$

Geometric Series

Let $\sequence {x_n}$ be a geometric sequence in $\R$:

$x_n = a r^n$ for $n = 0, 1, 2, \ldots$

Then the series defined as:

$\ds \sum_{n \mathop = 0}^\infty x_n = a + a r + a r^2 + \cdots + a r^n + \cdots$

is a geometric series.

Arithmetic-Geometric Progression

The term arithmetic-geometric progression is used to mean one of the following:

Arithmetic-Geometric Sequence

An arithmetic-geometric sequence is a sequence $\sequence {a_k}$ in $\R$ defined as:

$a_k = \paren {a_0 + k d} r^k$

for $k = 0, 1, 2, \ldots$

Thus its general form is:

$a_0, \paren {a_0 + d} r, \paren {a_0 + 2 d} r^2, \paren {a_0 + 3 d} r^3, \ldots$

Arithmetic-Geometric Series

An arithmetic-geometric series is a series whose underlying sequence is an arithmetic-geometric sequence:

\(\ds S_n\)

\(=\)

\(\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k\)

\(\ds \)

\(=\)

\(\ds a + \paren {a + d} r + \paren {a + 2 d} r^2 + \cdots + \paren {a + \paren {n - 1} d}r^{n-1}\)

Harmonic Progression

The term harmonic progression is used to mean one of the following:

Harmonic Sequence

A harmonic sequence is a sequence $\sequence {a_k}$ in $\R$ defined as:

$h_k = \dfrac 1 {a + k d}$

where:

$k \in \set {0, 1, 2, \ldots}$

$-\dfrac a d \notin \set {0, 1, 2, \ldots}$

Thus its general form is:

$\dfrac 1 a, \dfrac 1 {a + d}, \dfrac 1 {a + 2 d}, \dfrac 1 {a + 3 d}, \ldots$

Harmonic Series

The series defined as:

$\ds \sum_{n \mathop = 1}^\infty \frac 1 n = 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots$

is known as the harmonic series.