# Triangular Numbers Sequence

What is Triangular Number?

A triangular number Tn is a figurate number that can be represented in the form of an equilateral triangular grid of elements such that every subsequent row contains an element more than the previous one.

List Of Triangular Numbers:

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, and so on.

Formula:

Triangular numbers correspond to the first-degree case of Faulhaber’s formula.

$T_{n}=\sum_{k=1}^{n}k=1+2+3+…+n=\frac{n(n+1)}{2}=(\frac{n+1}{2})$

Where,

$\left ( \frac{n+1}{2} \right )$ is the binomial coefficient. It represents the number of distinct pairs that can be selected from N+1 objects.

Further, $\left ( \frac{n+1}{2} \right )$ can be expressed as $\frac{(n+1)!}{(n+1-2)!2!}$. This can be further simplified as $\frac{n(n+1)}{2}$

By the above formula, we can say that the sum of n natural numbers results in a triangular number, or we can also say that continued summation of natural numbers results in a triangular number. The sum of two consecutive natural numbers always results in a square number.

$T1+T2= 1+3=4=2^{2}$ and $T2+T3= 3+6=9=3^{2}$

All even perfect numbers are triangular numbers, and every alternate triangular number is a hexagonal number given by the formula:

$M_{P}2^{p-1}=\frac{M_{p}(M_{p}+1)}{2}=T_{Mp}$<

Where MP is a Mersenne prime.

For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

#### Practise This Question

The rational numbers, which are their own reciprocals are