Geometrical Progressions

Summary: A sequence in which each term is a constant multiple of the previous term; includes the sum formulas for finite progressions and infinite decreasing progressions.

Sources: chapter-1.3.11

Last updated: 2026-05-02

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Definition

A geometrical progression is a sequence $a,ab,a{b}^2,a{b}^3,\dots$ where $b$ is the constant ratio (also called the exponent). The $k$-th term is $a{b}^{k-1}$ and a progression of $n$ terms has last term $a{b}^{n-1}$.

Sum formulas

Finite progression (§514):

$s=\frac{a(b^n-1)}{b-1}$

Infinite decreasing progression, ratio $r=b/c<1$ (§520):

$s=\frac{a}{1-r}=\frac{ac}{c-b}$

Infinite alternating progression (§521):

$s=\frac{ac}{c+b}$

Relation to other concepts

Geometrical progressions are the multiplicative analogue of arithmetical-progressions, which have a constant difference rather than a constant ratio. The binomial-theorem series and the infinite-series arising from rational functions (see ch1.2.5-infinite-series) are special cases of geometric series. The sum formula for infinite decreasing progressions is used in ch1.3.12-infinite-decimal-fractions to convert repeating decimals to fractions, and in ch1.3.13-calculation-of-interest to sum the geometric series of compound-interest contributions.

Related pages

• arithmetical-progressions
• geometrical-ratio
• infinite-series
• repeating-decimals
• compound-interest
• ch1.3.11-geometrical-progressions