Geometric progression
- Note: a geometric series is just a series derived from the terms of a geometric progression
- Note: geometric sequence redirects here.
Definition
A geometric progression is any sequence, [ilmath](c_n)_{n\in\mathbb{N}_{\ge 1} } [/ilmath]^{[Note 1]} where each term is of the form:
- [ilmath]c_k\eq ar^{k-1} [/ilmath] for [ilmath]k\in[/ilmath][ilmath]\mathbb{N}_{\ge 1} [/ilmath]
Explicitly the sequence goes:
- [ilmath](a,\ ar,\ ar^2,\ ar^3,\ \ldots\ ,\underbrace{\ ar^{k-1},\ }_{k^\text{th}\text{ term} }\ldots)[/ilmath]
As such we can characterise any geometric progression as a pair of numbers:
- [ilmath]G\eq (a,r)\in\mathbb{R}^2[/ilmath], which we identify with a function:
- [ilmath]G:\mathbb{N}_{\ge 1}\rightarrow\mathbb{R} [/ilmath] by [ilmath]G:k\mapsto ar^{k-1} [/ilmath]
This is natural considering that a sequence is a function which maps each integer, [ilmath]k[/ilmath], to the [ilmath]k^\text{th} [/ilmath] term (as explained on the sequence page)
- See geometric series (the series from a geometric progression) for information on the sum of a geometric sequence, or a sub-sequence of it.
Canonical Geometric Progression
The canonical geometric progression will refer to [ilmath](1,r)[/ilmath], although any [ilmath](a,r)[/ilmath] such that [ilmath]a\neq 0[/ilmath] would do. As we now demonstrate:
- Let [ilmath](u_k)_{k\in\mathbb{N}_{\ge 1} } [/ilmath] be the sequence of the geometric progression [ilmath](1,r)[/ilmath], meaning:
- [ilmath](u_k)[/ilmath] is the sequence: [ilmath]1,\ r,\ r^2,\ r^3,\ \ldots,\ \underbrace{r^{k-1} }_{k^\text{th}\text{ term} },\ \ldots[/ilmath]
- Let [ilmath](a,r)[/ilmath] be any other geometric progression with the same ratio, [ilmath]r[/ilmath], denote its terms by the sequence [ilmath](v_k)_{k\in\mathbb{N}_{\ge 1} } [/ilmath]
- Then [ilmath](v_k)[/ilmath] is the sequence [ilmath]a,\ ar,\ ar^2,\ ar^3,\ \ldots,\ \underbrace{ar^{k-1} }_{k^\text{th}\text{ term} },\ \ldots[/ilmath]
- We see that [ilmath]v_k\eq a u_k[/ilmath] for each [ilmath]k\in\mathbb{N}_{\ge 1} [/ilmath]
- We abuse notation by writing [ilmath](v_k)_k\eq a(u_k)_k[/ilmath] or even [ilmath](a,r)\eq a(1,r)[/ilmath]
- We see that [ilmath]v_k\eq a u_k[/ilmath] for each [ilmath]k\in\mathbb{N}_{\ge 1} [/ilmath]
- Then [ilmath](v_k)[/ilmath] is the sequence [ilmath]a,\ ar,\ ar^2,\ ar^3,\ \ldots,\ \underbrace{ar^{k-1} }_{k^\text{th}\text{ term} },\ \ldots[/ilmath]
Formally, this shows: [ilmath]\forall (a,r)\in\mathbb{R}^2\exists b\in\mathbb{R}\big[b(1,r)\eq(a,r)\big][/ilmath] - namely [ilmath]b\eq a[/ilmath] itself.
As mentioned, we need not use [ilmath](1,r)[/ilmath] as our canonical progression, any non-zero value in place of [ilmath]1[/ilmath] would do, however [ilmath]1[/ilmath] is the natural choice over [ilmath]\sqrt{2} [/ilmath], 5, or even [ilmath]-1[/ilmath]
See also
- Geometric series
- Geometric distribution - a probability distribution with ties to geometric sequences.
- Arithmetic progression
Notes
- ↑ Notice the sequence goes:
- [ilmath]c_1,\ c_2,\ c_3,\ \ldots[/ilmath] (starting from [ilmath]k\eq 1[/ilmath]), not:
- [ilmath]c_0,\ c_1,\ c_2,\ \ldots[/ilmath]