Chapter 1.3.4 — Of the Summation of Arithmetical Progressions
Summary: Euler derives the formula for the sum of an arithmetical progression by pairing symmetric terms, and applies it to natural numbers, odd numbers, and a table of progressions starting at 1 with varying differences.
Sources: chapter-1.3.4
Last updated: 2026-04-30
Key observation: symmetric pairs (§413–414)
In an arithmetical progression with first term , last term , and common difference , any two terms equidistant from the ends — the -th from the front and the -th from the back — sum to :
The summation formula (§415–417)
Write the progression forward and then backward, and add column by column. Each column sums to , and there are columns, giving a total of . This total is twice the desired sum , so:
Equivalently: multiply the sum of the first and last term by half the number of terms (or multiply half their sum by the full number of terms).
Closed form in terms of (§420)
Substituting :
This is the general formula when the last term is not given directly.
Special cases
Sum of the first natural numbers (§418, 421):
Examples: ; clock strokes in 12 hours .
Sum of the first odd numbers (§422, ):
The partial sums of the odd-number progression are always perfect squares.
Table of sums starting at 1 with difference (§424):
| Sum | |
|---|---|
The pattern continues: for difference the sum is .
Applied problem: horse-and-nails (§419)
A horse is bought for nails: the first costs 5 pence, the second 8, the third 11, increasing by 3 each time, with 32 nails total. The last term is , and the total cost is pence ( pounds shillings pence).