Triangular number

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

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

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

For example, the $n^{th}$ triangle number is $1 +2+3 + 4............. +(n-1)+(n)$

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.

Pascal's Triangle:

The rather simple recursive definition can be easily found by noting that $T_{n} = 1 + 2 + \ldots + (n-1) + n = (1 + 2 + \ldots + n-1) + n = T_{n-1} + n$ .

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Triangular number