Geometric progression

Note: a geometric series is just a series derived from the terms of a geometric progression

Note: geometric sequence redirects here.

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Definition

A geometric progression is any sequence, $(c_n)_{n\in\mathbb{N}_{\ge 1} }$ where each term is of the form:

• $c_k\eq ar^{k-1}$ for $k\in$$\mathbb{N}_{\ge 1}$

Explicitly the sequence goes:

• $(a,\ ar,\ ar^2,\ ar^3,\ \ldots\ ,\underbrace{\ ar^{k-1},\ }_{k^\text{th}\text{ term} }\ldots)$

As such we can characterise any geometric progression as a pair of numbers:

• $G\eq (a,r)\in\mathbb{R}^2$, which we identify with a function:
  • $G:\mathbb{N}_{\ge 1}\rightarrow\mathbb{R}$ by $G:k\mapsto ar^{k-1}$

This is natural considering that a sequence is a function which maps each integer, $k$, to the $k^\text{th}$ 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 $(1,r)$, although any $(a,r)$ such that $a\neq 0$ would do. As we now demonstrate:

• Let $(u_k)_{k\in\mathbb{N}_{\ge 1} }$ be the sequence of the geometric progression $(1,r)$, meaning:
  • $(u_k)$ is the sequence: $1,\ r,\ r^2,\ r^3,\ \ldots,\ \underbrace{r^{k-1} }_{k^\text{th}\text{ term} },\ \ldots$
  • Let $(a,r)$ be any other geometric progression with the same ratio, $r$, denote its terms by the sequence $(v_k)_{k\in\mathbb{N}_{\ge 1} }$
    • Then $(v_k)$ is the sequence $a,\ ar,\ ar^2,\ ar^3,\ \ldots,\ \underbrace{ar^{k-1} }_{k^\text{th}\text{ term} },\ \ldots$
      • We see that $v_k\eq a u_k$ for each $k\in\mathbb{N}_{\ge 1}$
        • We abuse notation by writing $(v_k)_k\eq a(u_k)_k$ or even $(a,r)\eq a(1,r)$

Formally, this shows: $\forall (a,r)\in\mathbb{R}^2\exists b\in\mathbb{R}\big[b(1,r)\eq(a,r)\big]$ - namely $b\eq a$ itself.

As mentioned, we need not use $(1,r)$ as our canonical progression, any non-zero value in place of $1$ would do, however $1$ is the natural choice over $\sqrt{2}$, 5, or even $-1$

See also

• Geometric series
  • Geometric distribution - a probability distribution with ties to geometric sequences.
• Arithmetic progression
  • Arithmetic series