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A triangular number counts the dots that form an equilateral triangle — 1, 3, 6, 10, 15, and so on. This tool calculates the nth triangular number directly, or checks whether a given number is triangular and finds which n it corresponds to.
How it works
The nth triangular number is n(n+1)/2. To check if a number is triangular, this formula is solved in reverse using the quadratic formula: n = (−1 + √(1 + 8×number)) / 2 — if this comes out to a whole number, the original number is triangular.
- Enter mode.
- Enter n (if finding the nth triangular number).
- Enter number to check (if checking).
- Click Calculate to see your results.
Examples
Finding the 6th triangular number
T₆ = 6×7/2 = 21.
Checking if a number is triangular
21 is triangular (it's T₆), but 22 is not, since solving the reverse formula for 22 does not give a whole number.
Who should use it
- Combinatorics and number theory coursework.
- Recreational mathematics and pattern exploration (triangular number sequences).
Industry applications
- Mathematics education (number theory and combinatorics)
- Recreational mathematics
Advantages
- Handles both directions: finding T_n from n, and identifying n from a candidate triangular number.
- Shows the quadratic-formula reasoning for the reverse check.
Limitations
- Limited to n up to 1,000,000 for the forward calculation to keep results manageable.
Common mistakes to avoid
- Assuming every number close to a triangular number is itself triangular — the check requires an exact whole-number solution to the reverse formula, not just an approximate match.
Best practices
- When checking by hand whether a number is triangular, test whether 8× that number + 1 is a perfect square — it's a quick way to verify before doing the full quadratic solve.
Tips
- Triangular numbers and square numbers are closely related — every square number can be written as the sum of two consecutive triangular numbers.