A triangular number is a figurate number that can be represented in the form of an equilateral triangle arranged in a series or sequence. Triangular numbers count objects arranged in an equilateral triangle, and they are a type of figurate number, along with square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers starts with 1 and continues with 3, 6, 10, 15, 21, and so on.

To calculate any triangular number, we can use the formula xn = n(n+1)/2, where xn is the nth triangular number. For example, the 5th triangular number is x5 = 5(5+1)/2 = 15, and the 60th triangular number is x60 = 60(60+1)/2 = 1830.

Some interesting properties of triangular numbers include:

• The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and such numbers never end in 2, 4, 7, or 9.
• In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9.
• Every other triangular number is a hexagonal number.
• The sum of consecutive triangular numbers is a square number.

Triangular numbers have applications in various fields, including mathematics, physics, and computer science. For example, they can be used to calculate the number of dots in a triangular grid, the number of possible routes in a triangular lattice, and the number of elements in a triangular matrix.