Interesting Facts about Triangular Numbers (original) (raw)

Last Updated : 23 Jul, 2025

Triangular Numbers form a sequence of numbers that can be represented in the form of an equilateral triangle when arranged in a series. The triangular numbers list includes numbers 1, 3, 6, 10, 15, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, ****...**

Here are some interesting facts about Triangular Numbers.

• The nth triangular number is equal to sum of first n natural numbers, .i.e., n x (n + 1)/2 (see the proof here]
• From given two consecutive triangular numbers, we can find the next, by computing their difference + 1 + second. For example given 10 and 15, we can get the next as (15 - 10) + 1 + 15 = 21.
• The sum of two consecutive triangular numbers always gives a perfect square. For example 1 + 3 = 4, 3 + 6 = 9.
  We can simply prove it by taking sum of sum of n-th and (n+1)-th
  which is n x (n + 1)/2 + (n + 1) (n + 2)/2 = (n + 1)/2 [ n + n + 2] = (n + 1)2
• They are well known for their application in solving handshake problems
• The last digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9
• Triangular numbers are part of a larger family known as figurate numbers, which represent shapes like squares, pentagons, and hexagons.
• Sum of first n triangular numbers can be computed using a formula as n*(n+1)*(n+2) / 6. We can prove this formula by summing i*(i+1)/2 from i = 1 to n and then applying formulas of sum and sum of squares of first n natural numbers.
• Pascal's triangle is a triangular arrangement of numbers which are obtained in similar fashion as triangular numbers. Firstly, 1 is placed at the top. The numbers we get in subsequent steps is the addition of above two numbers. The triangular numbers appear in Pascal's triangle along the diagonal immediately below the first one, where the binomial coefficients are located:Tn = (n+1)C2
• There are many triangular numbers that are also square numbers. For example 36, 1225 etc. _All square triangular numbers are found from the recursion Sn = 34Sn−1 - Sn-2 + 2 with S0 = 0 and S1 = 1 and S1=1.'
• Square of nth triangular number is equal to sum of the cubes of integers 1 to n. This can be expressed as **Σ n k=1 **k 3 **= Sun of First n Natural Numbers or (Σ n k=1 k) 2
• Eight Times a Triangular Number Plus 1 is always a Perfect Square. We can show it as 8 x [ n x (n + 1)/2 ] + 1 is always a perfect square. The term simplifies to 4n2 + 4n + 1 which is (2n + 1)2 [A perfect square]
• The only triangular number in Fibonacci Series are 1, 3, 21 and 55.
• The last digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9
• The digital root of a nonzero triangular number is always 1, 3, 6, or 9
• The digit root repeats after every 9th Triangular Number
• If we sum reciprocals of triangular numbers to infinity, it converges to 2. In other words Sum(1/Tn) where n goes from 1 to infinity is 2.

Relationship with Figurative Numbers

Triangular Numbers are a subset of other figurate numbers such as square, pentagon or hexagonal numbers. They have a wide variety of relations with other figurate numbers as well. Some of them are listed below:

• Every Hexagonal Number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number
• Sum of any two consecutive triangular number is a square number. For example, 6+10=16, which is a square of 4.
• The positive difference between any two triangular number is a trapezoidal number. For instance, 10-3=7, which is a trapezoidal number****.**