Triangular number

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The triangular numbers are the numbers $T_n$ which are the sum of the first $n$ natural numbers from $1$ to $n$.

Definition

The $n^{th}$ triangular number is the sum of all natural numbers from one to $n$. That is, the $n^{th}$ triangle number is $1 +2+3 + 4............. +(n-1)+(n)$.

For example, the first few triangular numbers can be calculated by adding 1, 1+2, 1+2+3, ... etc.

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rowStart -= 0.5;
}
return 0;


}

for (int n=1; n<5; ++n) {

 real value= n*(n+1)/2;
draw_triangle((value+5,n),n);
label( (string) value, (value+5, -2));


} </asy>

Formula

Using the sum of an arithmetic series formula, a formula can be calculated for $T_n$:

$T_n =\sum_{k=1}^{n}k = 1 + 2 + \ldots + n = \frac{n(n+1)}2$

The formula for finding the $n^{th}$ triangular number can be written as $\dfrac{n(n+1)}{2}$.

It can also be expressed as the sum of the $n^{th}$ row in Pascal's Triangle and all the rows above it. Keep in mind that the triangle starts at Row 0.