Triangular Numbers
Numbers of dots forming triangles.
Key idea
Numbers of dots forming triangles.
In depth
The nth triangular number counts dots in a triangle with n rows: T(n) = 1 + 2 + ... + n = n(n+1)/2. Gauss found it by pairing the first and last terms.
Worked example
T(5) = 5×6/2 = 15 = 1+2+3+4+5. And T(100) = 5050.
Formula
Real-world uses
- Counting bowling pins or billiard balls arranged in triangles
- Seating patterns in a theatre
Common mistake
Using n(n+1) without dividing by 2.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Triangular Numbers?
Numbers of dots forming triangles. It is a topic at the intermediate level in Arithmetic & Number Sense.
What is the formula for Triangular Numbers?
The key formula is Tₙ = n(n+1)/2.
Where is Triangular Numbers used in real life?
Real-world examples: Counting bowling pins or billiard balls arranged in triangles; Seating patterns in a theatre.
What should I learn before Triangular Numbers?
Make sure you are comfortable with Perfect Numbers, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Triangular Numbers?
Using n(n+1) without dividing by 2.
How can I practise Triangular Numbers?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Triangular Numbers?
Move on to Figurate Numbers next.
Related intermediate topics in Arithmetic & Number Sense
- Long Multiplication – Multiply by each digit of the multiplier, shift by place value, then add partial products.
- Long Division – Divide, multiply, subtract and bring down repeatedly to get the quotient.
- Factors and Multiples – Factors divide a number exactly; multiples are its products with whole numbers.
- Divisibility Rules – Quick tests: by 2 last digit even, by 3 digit-sum divisible by 3, by 5 ends 0/5, by 9 digit-sum divisible by 9.
- Prime Numbers – A prime has exactly two positive divisors: 1 and itself.
- Composite Numbers – Numbers greater than 1 with more than two factors.