ch1.3.11-geometrical-progressions

Chapter XI – Of Geometrical Progressions

Summary: Defines geometrical progressions, derives the general sum formula, and extends it to infinite decreasing and alternating-sign progressions.

Sources: chapter-1.3.11

Last updated: 2026-05-02

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Definition and parameters

A geometrical progression is a series in which each term is obtained from the preceding one by multiplying by a fixed constant called the exponent or ratio $b$. With first term $a$ and ratio $b$ the general progression is

$a,ab,a{b}^2,a{b}^3,\dots ,a{b}^{n-1}$

Four parameters characterise any such progression (§507):

Symbol	Meaning
$a$	first term
$b$	ratio (exponent)
$n$	number of terms
$a{b}^{n-1}$	last term

Given any three of these, the fourth is determined. If $b>1$ the terms increase; if $b=1$ they are constant; if $b<1$ they decrease.

Sum of a finite geometrical progression

Let $s=a+ab+a{b}^2+\cdots +a{b}^{n-1}$. Multiplying by $b$ gives $bs=ab+a{b}^2+\cdots +a{b}^n$. Subtracting the first equation from the second yields $(b-1)s=a{b}^n-a$, hence (§514):

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

Equivalently: multiply the last term $a{b}^{n-1}$ by $b$, subtract the first term $a$, and divide by $b-1$.

Worked examples

• Seven terms, $a=3$, $b=2$: last term $=3\cdot 2^6=192$; sum $=(192\cdot 2-3)/(2-1)=381$ (§515).
• Six terms, $a=4$, $b=3/2$: progression $4,6,9,27/2,81/4,243/8$; sum $=83\frac{1}{8}$ (§516).
• The horse-nail problem (§511): 32 nails, price doubling from 1 penny; last term $2^{31}$; total $=2^{32}-1=4294967295$ pence $=17895697l.1s.3d.$

Sum of an infinite decreasing progression

When the ratio is a proper fraction $b/c$ with $c>b>0$, the terms decrease to zero and the infinite sum converges. Writing the sum as $s=a+\frac{ab}{c}+\frac{a{b}^2}{c^2}+\cdots$, multiplying by $b/c$ and subtracting gives $(1-b/c)s=a$, hence (§520):

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

Rule: divide the first term by $1$ minus the ratio.

Examples

First term $a$	Ratio	Sum
$1$	$1/2$	$2$
$1$	$1/3$	$3/2$
$2$	$3/4$	$8$
$3/10$	$1/10$	$1/3$
$9$	$1/10$	$10$ (i.e. $9.9\overline{9}=10$)

Alternating-sign infinite progression

When signs alternate, $s=a-\frac{ab}{c}+\frac{a{b}^2}{c^2}-\cdots$, multiplying by $b/c$ and adding gives $(1+b/c)s=a$, so (§521):

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

Example: $\frac{3}{5}-\frac{6}{25}+\frac{12}{125}-\cdots =\frac{3}{7}$.

Connection to repeating decimals

The series $\frac{3}{10}+\frac{3}{100}+\cdots$ is a geometric progression with $a=3/10$ and ratio $1/10$; its sum is $\frac{1}{3}$. This connects the theory of infinite geometric progressions directly to repeating-decimals and infinite-decimal-fractions (§523–524).

Related pages

• geometrical-progressions
• geometrical-ratio
• arithmetical-progressions
• infinite-series
• repeating-decimals
• ch1.3.12-infinite-decimal-fractions