Arithmetical Progressions

Summary: An arithmetical progression is a sequence of numbers with a constant common difference; four parameters (first term, last term, common difference, number of terms) fully characterize it, and the sum of any finite progression has a closed formula.

Sources: chapter-1.3.2, chapter-1.3.3, chapter-1.3.4

Last updated: 2026-04-30

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Definition

A sequence $a_1,a_2,a_3,\dots$ is an arithmetical progression if consecutive terms differ by a fixed constant $d$ (the common difference). The sequence is increasing if $d>0$ and decreasing if $d<0$.

The $n$-th term is:

$a_n=a\pm (n-1)d,$

where $a=a_1$ is the first term.

Four parameters

A finite progression is characterized by:

Symbol	Meaning
$a$	first term
$z$	last term
$d$	common difference
$n$	number of terms

The four formulas linking them (any three determine the fourth):

$z=a\pm (n-1)d,d=\frac{z-a}{n-1},n=\frac{z-a}{d}+1.$

For $n$ to be a valid (integer) number of terms, $z-a$ must be divisible by $d$.

Sum formula

$S=\frac{n(a+z)}{2}=na+\frac{n(n-1)d}{2}.$

Derived by pairing the $k$-th term from the front with the $k$-th from the back; each pair sums to $a+z$.

Key special cases

Progression	$a$	$d$	Sum of $n$ terms
Natural numbers	$1$	$1$	$\frac{n(n+1)}{2}$
Odd numbers	$1$	$2$	$n^2$
Starting at $1$, diff $d$	$1$	$d$	$n+\frac{n(n-1)d}{2}$

Connection to polygonal numbers

Partial sums of arithmetical progressions beginning at 1 produce polygonal-numbers: triangular ($d=1$), square ($d=2$), pentagonal ($d=3$), hexagonal ($d=4$), and so on.

Related pages

• arithmetical-proportion
• ratios
• polygonal-numbers
• ch1.3.3-arithmetical-progressions
• ch1.3.4-summation-arithmetical-progressions
• infinite-series