A triangular number is a number that can be represented in the form of a triangular grid of points where the first row contains a single point and each subsequent row contains one more point than the previous one.
Mathematically, the nth triangular number is the sum of all positive integers less than or equal to n. The sequence of triangular numbers starts with: 1, 3, 6, 10, 15, 21, 28, 36, 45, …
The formula for the nth triangular number Tn is: Tn = n(n+1)/2
Simple Examples
- The 1st triangular number is 1
- The 2nd triangular number is 1 + 2 = 3
- The 3rd triangular number is 1 + 2 + 3 = 6
- The 4th triangular number is 1 + 2 + 3 + 4 = 10
- The 5th triangular number is 1 + 2 + 3 + 4 + 5 = 15
Real World Examples
-
Bowling pins: Bowling pins are arranged in a triangular pattern with 1 pin in the first row, 2 pins in the second row, 3 in the third, and 4 in the fourth row. The total number of pins (10) is the 4th triangular number.
-
Pool balls: The number of balls in a triangular rack used to set up a game of 8-ball or straight pool is the 5th triangular number: 15.
-
Stadium seating: Many stadiums have seating arranged in a triangular pattern, with each subsequent row having one more seat than the previous row. The total number of seats is a triangular number.
-
Stacking objects: When stacking identical objects like cans in a triangular pyramid, the total number of objects used will be a triangular number based on the number of rows.
In summary, triangular numbers pop up frequently in scenarios involving triangular patterns, summing sequences, or counting combinations in real life and in mathematical problems.
