Triangular Numbers
Compute T(n)=n(n+1)/2 for any n instantly. Triangular numbers calculator finds the nth term and sum of first n positive integers for math students and teachers.
About This Calculator
Our Triangular Numbers Calculator computes the nth triangular number T(n) for any non-negative integer n. Triangular numbers are a sequence of numbers that can be arranged in the shape of an equilateral triangle. The nth triangular number equals the sum of the first n positive integers, defined by the formula T(n) = n x (n+1) / 2.
This free online calculator is ideal for students learning number sequences and patterns, teachers demonstrating figurate numbers, math enthusiasts exploring OEIS sequence A000217, and anyone who needs to quickly compute triangular numbers without manual addition. Simply enter the position n to get the corresponding triangular number instantly.
The triangular numbers sequence begins: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666 and continues infinitely. These numbers have connections to binomial coefficients, perfect numbers, hexagonal numbers, and the handshake problem in combinatorics.
The graphical proof of the triangular numbers formula involves arranging dots in a right triangle, duplicating and flipping it to form a rectangle of dimensions n by n+1. Since the rectangle contains n x (n+1) dots and the triangle is exactly half, we get T(n) = n(n+1)/2. This elegant geometric demonstration helps visualize why the formula works.
Applications
Triangular numbers appear in the handshake problem: in a group of n people, the total number of handshakes is T(n-1). They also count connections in fully connected networks, arrangements of objects in triangular formations (like bowling pins or billiard balls), and seating arrangements. In number theory, every even perfect number is a triangular number, and alternating triangular numbers correspond to hexagonal numbers.
Frequently Asked Questions
What is a triangular number?
A triangular number is a number that can be arranged in the shape of an equilateral triangle. The nth triangular number T(n) equals the sum of the first n positive integers: 1 + 2 + 3 + ... + n = n(n+1)/2. The sequence starts 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105 and continues infinitely.
How do you calculate triangular numbers?
Use the formula T(n) = n x (n+1) / 2. For example, to find the 10th triangular number: T(10) = 10 x 11 / 2 = 55. Alternatively, you can add the first 10 positive integers: 1+2+3+4+5+6+7+8+9+10 = 55. Simply enter n into this calculator to get an instant result.
Is zero a triangular number?
Yes, by convention zero is considered the 0th triangular number. It satisfies T(0) = 0 x 1 / 2 = 0 and is consistent with the sequence definition as the sum of zero terms. Many mathematical references list 0 as the first entry in OEIS A000217.
What are triangular numbers used for in real life?
Triangular numbers appear in the handshake problem (the number of handshakes in a group of n people is T(n-1)), fully connected network node counts, bowling pin arrangements, and seating arrangements. They also relate to perfect numbers, hexagonal numbers, and binomial coefficients in combinatorics.
What is the relationship between triangular and square numbers?
The sum of any two consecutive triangular numbers equals a square number. For example, T(3) = 6 and T(4) = 10, and 6 + 10 = 16 = 4 squared. This holds for any n: T(n) + T(n+1) = (n+1) squared.
Are all triangular numbers positive?
Triangular numbers are non-negative by definition. T(0) = 0, T(1) = 1, T(2) = 3, and the sequence grows positively from there. Since the formula n(n+1)/2 uses multiplication of non-negative integers, the result is always a non-negative integer.
How fast do triangular numbers grow?
Triangular numbers grow quadratically with n. T(10) = 55, T(100) = 5,050, T(1,000) = 500,500, and T(10,000) = 50,005,000. The growth rate is approximately half of n squared since T(n) is roughly n squared divided by 2 for large n.