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Triangular Number Calculator

Work out triangular number instantly with clear inputs, formula shown and shareable results.

About Triangular Number Calculator

The nth triangular number counts dots in a triangular array and equals n(n + 1)/2. A value v is triangular exactly when 8v + 1 is a perfect square.

Formulas

Triangular number

Tₙ = n(n + 1) / 2

Triangular test

v is triangular ⇔ 8v + 1 is a perfect square

Worked example

With the default values below, Triangular Number Calculator returns nth triangular number of 210. Change any input above and every figure here is recalculated in your browser.

Inputs used

Index naccepts 1–100,000,000
20
Test whether this value is triangularminimum 1
210

Results

nth triangular number
210
Next triangular number
231
Your test value
Yes, it is T20
Equals 1 + 2 + … + n
210

Step by step

  1. 1Values used
  2. 2Triangular number
  3. 3Triangular test
  4. 4nth triangular number= 210
  5. 5Next triangular number= 231
  6. 6Your test value= Yes, it is T20
  7. 7Equals 1 + 2 + … + n= 210

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is T₂₀?

20 × 21 / 2 = 210, which is also the number of handshakes among 21 people.

Which numbers are both square and triangular?

1, 36, 1225, 41616 and so on — an infinite family generated by the Pell equation.

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