Calculate Triangular Number and Triangular root
Calculator for the nth triangular number and the triangular root of a number. A triangular number is the sum of consecutive natural numbers, similar to stacking equal-sized objects on top of each other, with each row containing one more object, so the stack becomes triangular. The triangular root is the reverse calculation.
Example: For n=4, the triangular number is 10. There are ten spheres in four rows, with the top row containing one sphere, the second row containing two, the third row containing three, and the bottom row containing four spheres. The triangular root of 10 is therefore 4.
The n-th triangular number can be calculated using the Gaussian sum formula.
If you want to display the sequence of triangular numbers, i.e. all triangular numbers up to a certain number in a row, you can enter the formula i*(i+1)/2 into the calculator for the sequence.
The inversion of the Gaussian sum formula is done via the quadratic equation and leads to
There are two results here, with the first usually being the relevant result. The second result is most often negative.
Unlike the square root, the triangular root has no such relationship between length and area. A square with side length a has an area a², while one with area A has an edge length √A. Calculating the area of triangles, however, cannot be done using the triangular number and the triangular root. Triangular numbers have been known since ancient times and play a role in some mathematical problems. For example, if n people meet and everyone shakes hands, then the total number of handshakes will be the n-1. triangular number. This is a simple problem from the mathematical field of combinatorics.
