Triangular Numbers Calculator
Calculate triangular numbers and verify if a number is triangular with formula explanations
Calculate Triangular Numbers
Enter a positive integer to find its triangular number
Triangular Number Result
Example Calculations
Find 6th Triangular Number
T₆ = 6 × (6 + 1) / 2
T₆ = 6 × 7 / 2
T₆ = 21
Sum: 1+2+3+4+5+6 = 21
Check if 10 is Triangular
n² + n - 20 = 0
n = (-1 + √81) / 2
n = (-1 + 9) / 2 = 4
Yes! T₄ = 10
First 20 Triangular Numbers
Properties
Every triangular number is the sum of consecutive integers from 1 to n
Sum of two consecutive triangular numbers is a perfect square
Formula: Tn = n(n+1)/2 = C(n+1,2)
Used in handshake problems and network connections
Quick Tips
Triangular numbers grow quadratically
Every even perfect number is triangular
Can be visualized as triangular dot patterns
Related to binomial coefficients
Understanding Triangular Numbers
What are Triangular Numbers?
Triangular numbers are a sequence of numbers that can be arranged in the shape of an equilateral triangle. Each triangular number represents the sum of consecutive positive integers starting from 1.
The Formula
Tn = n × (n + 1) / 2
This formula gives us the nth triangular number, where n is the position in the sequence.
Applications
- •Handshake Problem: Number of handshakes in a group of n people
- •Network Connections: Connections in a fully connected network
- •Combinatorics: Number of ways to choose 2 items from n+1 items
- •Geometry: Patterns in dot arrangements and polygonal numbers
Visual Pattern
T₁ = 1 dot (•)
T₂ = 3 dots (••• arranged in triangle)
T₃ = 6 dots (•••••• arranged in triangle)
And so on...
Interesting Properties
Sum Property
Tn + Tn+1 = (n+1)²
The sum of two consecutive triangular numbers is always a perfect square.
Binomial Connection
Tn = C(n+1, 2)
Triangular numbers are binomial coefficients "n+1 choose 2".