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Triangular Number Calculator

Find the nth triangular number.

How it is calculated

The nth triangular number counts the dots in a triangle with n dots on its base, which equals the sum 1 + 2 + … + n. The closed form is Tn = n × (n + 1) / 2, the trick attributed to pairing the first and last numbers of the sum. Enter n as a whole number of 0 or more; decimals are cut to the integer part and negative numbers return no result. The tool gives T0 = 0, T1 = 1, T2 = 3, T3 = 6 and so on. The same number counts handshakes among n + 1 people, since each pair shakes once.

Worked example

Find the 10th triangular number. Using the formula, T10 = 10 × 11 / 2 = 110 / 2 = 55. Check by adding: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. As a picture, a triangle with 10 dots in the bottom row, 9 in the next and so on down to 1 contains 55 dots in total.

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Common mistakes

  • Computing n² / 2, which ignores the +1 and gives 50 for n = 10.
  • Starting the count at 0 or 1 inconsistently; T1 = 1 is the first triangular number, T0 = 0.
  • Confusing triangular numbers with square numbers or Fibonacci numbers.
  • Entering a decimal such as 10.9, which is cut to 10, not rounded to 11.
  • Using n for the number of people in a handshake problem; for n people use n − 1 here.

About this calculator

This calculator finds the nth triangular number using the formula T(n) = n(n+1)/2 — the sum of all whole numbers from 1 up to n, and the same count you'd get by stacking dots into a triangle with n dots on each side. T(1)=1, T(2)=3, T(3)=6, and so on.

Math students use it for number theory and combinatorics homework, and programmers use triangular numbers when counting handshakes in a group, pairs in a round-robin tournament, or the number of cells below a diagonal in a grid.

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