1. What Are Triangular Numbers?

Triangular numbers are a sequence of numbers where each number represents a triangle with dots. The nth triangular number is the number of dots composing an equilateral triangle with n dots on a side.

2. How the Formula Works

The formula for triangular numbers is:

Where:

• \( T_n \) — The nth triangular number
• \( n \) — The position in the sequence (starting from 1)

Explanation: The formula sums all integers from 1 to n. For example, the 4th triangular number is 1+2+3+4 = 10.

3. Applications of Triangular Numbers

Details: Triangular numbers appear in various mathematical contexts including combinatorics, number theory, and geometry. They're used in calculating handshakes, tournament schedules, and more.

4. Using the Calculator

Tips: Enter any non-negative integer n to calculate the nth triangular number. The calculator will compute T_n = n(n+1)/2.

5. Frequently Asked Questions (FAQ)

Q1: What are the first 10 triangular numbers?
A: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55

Q2: Can n be zero?
A: Yes, T₀ = 0 by definition.

Q3: How are triangular numbers related to square numbers?
A: The sum of two consecutive triangular numbers is always a square number (T_n-1 + T_n = n²).

Q4: What's the largest triangular number this calculator can handle?
A: It depends on your system's integer limits, but typically up to n = 2,147,483,647.

Q5: Are triangular numbers used in real-world applications?
A: Yes, they're used in network design, combinatorial mathematics, and even in some financial calculations.