ch1.3.3-arithmetical-progressions

Chapter 1.3.3 — Of Arithmetical Progressions

Summary: Euler formalizes arithmetical progressions and derives the four fundamental formulas linking first term, last term, common difference, and number of terms — any three of which determine the fourth.

Sources: chapter-1.3.3

Last updated: 2026-04-30

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Definition and terminology (§402–404)

An arithmetical progression is a sequence of numbers that increases or decreases by a fixed quantity called the common difference (or simply the difference).

Examples:

• Natural numbers $1,2,3,4,\dots$ (difference $=1$, increasing)
• $25,22,19,16,13,\dots$ (difference $=3$, decreasing)
• First term $2$, difference $3$: $2,5,8,11,14,17,20,23,26,29,\dots$

The position of each term is tracked by an index written above the sequence. The $n$-th term has index $n$.

General term formula (§405)

Let $a$ = first term, $d$ = common difference. The $n$-th term is:

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

with $+$ for an increasing and $-$ for a decreasing progression. In particular the tenth term is $a\pm 9d$ and the hundredth term is $a\pm 99d$.

The four parameters (§406–411)

Four quantities characterize a finite arithmetical progression:

1. First term $a$
2. Last term $z$
3. Common difference $d$
4. Number of terms $n$

Any three of these determine the fourth:

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

Remark on integrality (§410): Since $n$ must be a positive integer, the formula $n=\frac{z-a}{d}+1$ requires $z-a$ to be exactly divisible by $d$; otherwise the problem has no solution.

Worked examples (§408–409)

Finding $d$: progression of 9 terms, first term $2$, last term $26$:

$d=\frac{26-2}{9-1}=\frac{24}{8}=3\Rightarrow 2,5,8,11,14,17,20,23,26.$

Fractional difference: first term $1$, last term $2$, 10 terms:

$d=\frac{2-1}{10-1}=\frac{1}{9}\Rightarrow 1,1\frac{1}{9},1\frac{2}{9},\dots ,2.$

Finding $n$: first term $4$, last $100$, difference $12$:

$n=\frac{100-4}{12}+1=8+1=9.$

Related pages

• arithmetical-progressions
• ch1.3.2-arithmetical-proportion
• ch1.3.4-summation-arithmetical-progressions
• ratios