**Triangular Numbers:**

A figurate number or a figural number is a number which can be expressed using a regular geometrical arrangement of points which are equally spaced. The number is also known as polygonal number if the geometric arrangement forms a regular polygon such as triangle, square, pentagon, etc.

A triangular number \(T_n\)

Fig. 1 shown below represents \(T_n\)

In simple terms, a triangular number is the number obtained by adding all natural numbers less than or equal to given positive integer \(n\)

\(T_n\)

n+1\cr

2

\end{matrix}\right) \)

\(\left(\begin{matrix}

n\cr

k

\end{matrix}\right)\)

n+1\cr

2

\end{matrix}\right)\)

Thus, the sequence of triangular numbers starting from \(n~=~1\)

\(1,3,6,10,15,21,28,36,45……….\)

Therefore, the continued summation of natural numbers results in a sequence of triangular numbers. These numbers are closely interrelated with other figurate numbers. The sum of two consecutive triangular numbers gives a square number.

\(T_1 + T_2~=~1 + 3~=~4~=~2^2\)

\(T_2 + T_3~=~3 + 6~=~9~=~3^2\)

\(T_3 + T_4~=~6 + 10~=~16~=~4^2\)

….

\(T_{n-1} + T_n\)

Also, all even perfect numbers are triangular i.e. \(T_p\)

Thus study of figurate numbers and triangular numbers is very interesting and appealing and it has got various real life applications such as fully interconnected network of n computing devices requiring \(T_{n-1}\)