# Definition: Triangle Numbers

If we put building bricks into rows, beginning from $$1$$ brick in the first row, $$2$$ bricks in the second, and so on, the total number of bricks is the $$n$$-th triangle number $$\Delta_n$$:

|$\begin{array}{c} \fbox x \end{array}$|$\begin{array}{cc} \fbox x\\ \fbox x&\fbox x \end{array}$|$\begin{array}{ccc} \fbox x\\ \fbox x&\fbox x\\ \fbox x&\fbox x&\fbox x \end{array}$|$\begin{array}{cccc} \fbox x\\ \fbox x&\fbox x\\ \fbox x&\fbox x&\fbox x\\ \fbox x&\fbox x&\fbox x&\fbox x \end{array}$|$\begin{array}{ccccc} \fbox x\\ \fbox x&\fbox x\\ \fbox x&\fbox x&\fbox x\\ \fbox x&\fbox x&\fbox x&\fbox x\\ \fbox x&\fbox x&\fbox x&\fbox x&\fbox x \end{array}$|$$\cdots$$|

$$\Delta_1:=1$$ $$\Delta_2:=3$$ $$\Delta_3:=6$$ $$\Delta_4:=10$$ $$\Delta_5:=15$$ $$\cdots$$

The general formula for the $$n$$-th triangle number is

$\Delta_n=\sum_{k=1}^n n=1+2+\cdots+n=\frac {(n+1)n}{2},$

which can easily be proven as a special case of the sum of arithmetic progression.

The sequence of triangle numbers begins with $$1$$, $$3$$, $$6$$, $$10$$, $$15$$, $$21$$, $$28$$, $$36$$, ...

Explanations: 1

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