A triangle made of rows
Triangular numbers are totals built from rows of 1, then 2, then 3, and so on. The first positive triangular numbers are 1, 3, 6, 10, 15, 21.
Arrange six counters in three rows: one at the top, two beneath it, then three beneath those. The total is 1 + 2 + 3 = 6. The growing rows give the numbers their name.
Cambridge’s NRICH lists the triangular totals. The small examples here let you check how they grow.
The next row gives the next total
Add a fourth row with four counters: 6 + 4 = 10. A fifth row adds five: 10 + 5 = 15.
After five rows, the next row has six counters. That gives 15 + 6 = 21. You need the next row’s size, rather than adding the same amount every time.
Read the gaps as well as the totals
Look at 1, 3, 6, 10, 15. The jumps are 2, 3, 4, 5. Each new jump is one larger because the next row is one counter longer.
Starting with one counter and this rule fixes what follows. A short list by itself might fit several rules, so keep the row instruction alongside the numbers.
Compare the rule with Fibonacci
In the Fibonacci sequence, you add the previous two terms. For triangular numbers, you add the next row size.
Both sequences can contain 21, but a shared number does not make their rules identical. To continue a pattern, ask what you are supposed to add, then check every step.