**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 \(T_n\), is a figurate number representable in form of triangular grid of points in such a manner that each succeeding row consists of one element more than preceding one.

Fig. 1 shown below represents \(T_n\) where \(T_1\) = 1, \(T_2\) = 3, \(T_3\) = 6 and so on.

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

\(T_n\) = \(∑_{k=1}^n~k~=~\frac{1}{2}n(n+1)\) = \(\left(\begin{matrix}

n+1\cr

2

\end{matrix}\right) \)

\(\left(\begin{matrix}

n\cr

k

\end{matrix}\right)\) = \(nC_k\) = \(C(n,k)\) = \(\frac{n!}{(n – k)! k!}\) is the binomial coefficient. It represents the number of ways in which \(k\) unordered outcomes can be selected from a total of \(n\) possibilities. Here,\(\left(\begin{matrix}

n+1\cr

2

\end{matrix}\right)\) is the binomial coefficient which indicates the distinct pairs that can be selected from \(n+1\) objects.

Thus, the sequence of triangular numbers starting from \(n~=~1\) can be given as:

\(1,3,6,10,15,21,28,36,45……….\)

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.

\(T_1 + T_2~=~1 + 3~=~4~=~2^2\) \(T_2 + T_3~=~3 + 6~=~9~=~3^2\)\(T_3 + T_4~=~6 + 10~=~16~=~4^2\)

….

\(T_{n-1} + T_n\) = \((T_n – T_{n-1})^2~=~n^2\)

Also, all even perfect numbers are triangular i.e. \(T_p\) where \(p\) is prime. It is also very interesting to note that every alternate triangular number is a hexagonal number i.e. \(1,6,15,28……\)

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 \(T_{n-1}\) cables, calculation depreciation of an asset and a variety of other applications.