Nth Triangular Number Generator

About this calculator

A triangular number counts how many dots it takes to arrange objects into an equilateral triangle — 1, then 3, then 6, then 10 — and is given by the closed-form formula T(n) = n(n+1)/2, the same sum 1+2+3+...+n that a young Gauss is famously said to have shortcut by pairing the first and last terms. Triangular numbers show up throughout combinatorics (T(n) equals the number of ways to choose 2 items from n+1, and the number of unique handshakes among n+1 people) and recreational math (they're the numbers a bowling-pin or pool-ball rack naturally forms).

This calculator takes an index n and returns T(n) along with the previous and next triangular numbers in the sequence, so you can see the pattern around your chosen index. Students learning summation formulas and mathematical induction, puzzle and combinatorics enthusiasts, and programmers checking a closed-form result against a loop-based sum use it to quickly get or verify a triangular number without hand-summing a long series.