Triangular Numbers Calculator - nth Value, Sum, and T_n

A triangular numbers calculator finds the nth triangular number for any non-negative integer n in one step, returning the closed-form value n(n+1)/2 alongside the sum 1+2+...+n, the previous triangular number, and the next one.

• • Class homework and number theory: Verify hand calculations for n(n+1)/2, the Gauss sum, and binomial-coefficient problems that reduce to C(n+1, 2).
• • Programming and algorithm work: Generate figurate-number test data, validate integer recurrences, and confirm that T_n is the sum of the first n positive integers.
• • Combinatorics sanity checks: Confirm the well-known identity C(n+1, 2) = n(n+1)/2 when counting two-element subsets, handshakes, or edges in a complete graph.
• • Puzzle and contest answer checking: Cross-check a candidate triangular number, including the famous 1+2+...+100 = 5050 puzzle, against the closed form in a single step.

The triangular numbers count the dots that form an equilateral triangle of side n, so 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... is the same as 1+2+3+...+n. The closed form n(n+1)/2, often attributed to Gauss, is what the calculator evaluates directly, then cross-checks against the sum from 1 to n. The previous and next triangular numbers are returned alongside so you can see the gap your value sits in.

If you want the related arithmetic-sequence view, Arithmetic Sequence Calculator takes a starting term, common difference, and term index and returns any term of an arithmetic progression.

The calculator reads n, multiplies n by n+1, divides by 2 to get the closed-form triangular number, and cross-checks the result by summing 1 through n. If the input is negative or non-integer, the result is zeroed and the verdict flags the input as not in the triangular sequence.

• n: Position in the triangular sequence, a non-negative integer.
• T(n): The nth triangular number, computed as n(n+1)/2.
• 1+2+...+n: Sum of the first n positive integers, used as an independent cross-check.
• T(n-1): Previous triangular number, equal to (n-1)*n/2, or 0 when n = 0.
• T(n+1): Next triangular number, equal to (n+1)*(n+2)/2.

For every non-negative integer n, the closed form and the sum return the same integer, so the two results agree by construction. The previous and next values are computed from the same closed form rather than by stepping from T_n, which keeps the surrounding values exact for any n in the supported range.

Negative or non-integer n is rejected because the triangular sequence is defined for whole-number positions only. The calculator returns T_n = 0, zeroes the surrounding values, and marks the input as not in the sequence, matching the convention in number theory and integer combinatorics.

According to Wikipedia (Triangular number), triangular numbers equal the sum of the first n positive integers, follow the closed form n(n+1)/2, and match the binomial coefficient C(n+1, 2)

When the sequence you care about is the Fibonacci numbers rather than the triangular sequence, Fibonacci Calculator returns any F_n along with the previous two values and a sum of the first n terms.

Four ideas show up every time you talk about triangular numbers, and they cover everything the calculator reports.

These four ideas cover the closed form, the dot-triangle picture, the binomial connection, and the parity pattern. The sequence list below shows how the values grow and makes the parity pattern visible at a glance.

For the parallel integer-power test on the n^2 sequence, Perfect Square Calculator tests whether a value is a perfect square and lists the previous and next squares around your input.

Five short steps take you from a raw integer n to the closed-form value, the sum, and the surrounding triangular numbers.

1. 1 Enter the position n: Type a non-negative integer from 0 to 99999. The default n = 10 returns T_10 = 55.
2. 2 Read the nth triangular number: The headline result T_n is the closed-form value n(n+1)/2 at your n. For n = 10 the calculator returns 55.
3. 3 Check the sum 1+2+...+n: The sum field is the same value recomputed by adding the first n positive integers, so it should match T_n exactly. A mismatch means the closed form and the sum disagree, which would be a calculator bug.
4. 4 Read the previous and next triangular numbers: T_(n-1) and T_(n+1) are returned alongside T_n, so you can see the gap your value sits in. For n = 10 the previous is 45 and the next is 66.
5. 5 Try a large n to test the formula: Enter n = 100 to see T_100 = 5050, the classic Gauss sum. The closed form stays exact for any n in the supported range.

Try n = 100: closed form 100 * 101 / 2 = 5050, sum 1+2+...+100 = 5050, previous 4950, next 5151. The two results match exactly, confirming that n(n+1)/2 is the right closed form for the sum.

If you want to test whether the nth triangular number itself is prime, Prime Number Checker returns the full prime check, nearest primes, and the prime factorization of your input.

Five benefits make the calculator more useful than the formula by hand.

• • Closed form plus sum cross-check: T_n = n(n+1)/2 and the sum 1+2+...+n are both returned, so you can confirm they match. A mismatch would be a bug; agreement confirms the formula is being evaluated correctly.
• • Surrounding triangular numbers included: T_(n-1) and T_(n+1) are shown next to T_n, so you can see the gap your value sits in without writing out the sequence by hand.
• • Exact integer arithmetic for any n: The closed form returns the integer value of T_n with no rounding, even for large n like 1000, where T_1000 = 500500 exactly.
• • Negative and non-integer input handling: Negative or non-integer n is rejected with a clear note, matching the convention in number theory that triangular numbers are defined for n >= 0 only.
• • First 50 triangular numbers reference: The reference table below the calculator lists T_n for n = 0 to 50, so you can read off the parity pattern and the gaps between consecutive values.

Each benefit maps to a real workflow: homework, contest prep, combinatorics, and sequence work in programming.

For the multiplicative counterpart to the additive triangular sequence, Geometric Sequence Calculator takes a starting term, common ratio, and term index and returns any term of a geometric progression.

Three factors and two practical limits decide what the calculator returns.

• • The calculator handles one n at a time, so it does not plot the sequence as a graph. Use the reference table below for a longer view of T_n at small n.
• • The cross-check sum 1+2+...+n uses a loop, so the running time grows with n. The closed form n(n+1)/2 is what the calculator relies on for large n, and the sum is a sanity check rather than the primary computation.

The closed form and the binomial-coefficient form C(n+1, 2) are the same identity in two guises, so the answer is consistent with the way combinatorial problems count two-element subsets.

According to Wolfram MathWorld (Triangular Number), triangular numbers satisfy the recurrence T_{n+1} = T_n + n + 1, with T_0 = 0, T_1 = 1, T_2 = 3, T_3 = 6, and closed form T_n = n(n+1)/2

According to OEIS A000217, the triangular-number sequence is 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... with formula a(n) = n(n+1)/2

When the integer power you want to test is 3 rather than 2, Perfect Cube Calculator checks whether a value can be written as a^3 and lists the first 100 perfect cubes.