NUMBER LOGIC / JD · 2026.10.19
Triangular Numbers and the 1 + 2 + … + n Pattern
How to count stacked dots and derive a simple formula.
Build a triangle
One dot, then a row of two, then three and so on gives totals 1, 3, 6, 10, 15 and 21. The nth triangular number is the sum of the first n positive integers.
Deriving the formula
Pair the sum 1 + 2 + … + n with the same sum written backwards. Each pair totals n + 1, and there are n pairs, so twice the sum is n(n + 1). Thus Tₙ = n(n + 1)/2.
Examples and caveats
T₁₀ = 55 and T₂₀ = 210. A large number of divisors does not make an integer triangular; the defining property is its position in this exact sum sequence.
Try inverse reasoning
To test whether 45 is triangular, solve n(n + 1)/2 = 45. Since 9 × 10 / 2 = 45, it is T₉. Compare square numbers and triangular numbers to see different figurate patterns.
The calculations are mathematical facts. Interpretations attached to popular number symbolism are a separate, non-scientific subject.