Growing sums

Triangular numbers as cumulative sums

See how adding one longer row at a time produces 1, 3, 6, 10, 21, and 55.

A triangular number counts the objects that can be arranged in rows of 1, 2, 3, and so on. The nth triangular number is the sum of the counting numbers from 1 through n. Number Atlas uses additive fixed-value symbols rather than dot triangles, so this guide connects each reviewed visual to its triangular sum.

The rule

Add one longer row

Number-symbol visualization of 21 from the reviewed Numerosidad Number Atlas collection. — Curated collection

21

Add the counting numbers from 1 through n. 6 reviewed triangular values.
Explore the reviewed page for 21

The first triangular number is 1. Adding a second row of 2 gives 3; adding a third row of 3 gives 6; adding a fourth row of 4 gives 10.

The compact formula T(n) = n(n + 1) / 2 produces the same values as the cumulative sum 1 + 2 + ... + n.

This curated set skips triangular values that do not yet have reviewed permanent pages. The sequence itself is continuous; the page collection is selective.

RuleT(n) = 1 + 2 + ... + n = n(n + 1) / 2

Reviewed sums

Triangular numbers already in the permanent collection

The row count identifies the position in the triangular sequence. Compare the sum expression with the fixed-value symbol visualization on each reviewed page.

Reviewed triangular pages

The selected values show early growth and two later sequence intersections.

1T(1) = 1 3T(2) = 1 + 2 = 3 6T(3) = 1 + 2 + 3 = 6 10T(4) = 1 + 2 + 3 + 4 = 10 21T(6) = 21 55T(10) = 55

Look closely

Patterns worth comparing

The gap grows by one

The differences between consecutive triangular numbers are 2, 3, 4, 5, and onward. Every new triangular number adds the next counting number.

Standard decimal notation can cross a boundary

Adding 4 to 6 produces 10, where ten ones regroup into one ten. Later sums distribute across both tens and ones.

Sequence intersections

1, 3, 21, and 55 are also Fibonacci values in the reviewed collection; 55 is also palindromic.

Use the collection

Connect sums to symbol values

01
Build the sum
Choose a reviewed triangular value and write the consecutive counting-number sum that produces it.
02
Find the next row
Add the next counting number and predict the following triangular value, even when that value has no permanent page.
03
Compare representations
Explain how a dot triangle and a fixed-value symbol visualization represent the same total while organizing it for different purposes.

Keep exploring