Triangular Numbers

Pythagoras, the Greek mathematician is known as ‘The father of numbers.’ He explored the mathematical relationship with figurate numbers.
Pythagoras discovered triangular numbers. Triangular numbers are also called Triangle numbers. Pythagoras studied the triangular numbers under figurate numbers in geometry.
First natural number is 1.
Sum of the first two consecutive natural numbers is 1+2 =3
Sum of the first three natural numbers is 1+2+3=6
Sum of the first four natural numbers is 1+2+3+4=10
Sum of the first five natural numbers is 1+2+3+4+5=15
1, 3, 6, 10, 15…are triangular numbers. A    triangle number is the sum of first n natural numbers.
Fredric Gauss discovered the formula for nth triangular number.

Nth triangular number=n (n+1)/2.
For example: 6 th triangular number= 6(6+1)/2=  6(7)/2=  21

Let’s try one more example.
36 th triangular number =36 (36+1) /2
                                         =36 (37) /2= 666.
             666 is called the beast number.

• Every triangular number is a natural number.

• Every triangular number except 1 is divisible by 3 or when divided by 9 the remainder is 1.

• Triangular numbers can never end in 2, 4, 7, or 9.

• Eight times of any triangular number plus 1 is always a perfect square number.

• Nine times of any triangular number plus 1 is always a triangular number.

• Sum of any two consecutive triangular numbers is always a perfect square number.

• Difference between two consecutive squares of any triangular numbers is always a perfect cube of a small triangular number.

• Some triangular numbers are there which reversals are also triangular numbers. The examples are 1,3,6,10,55,66,120,153,171,190 300,595,630,666,820.3003,etc.