Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The first 100 terms sequence of triangular numbers, starting with the 0th triangular number, are

(sequence A000217 in the OEIS)

The triangular numbers are given by the following explicit formulas:

where ${\displaystyle \textstyle {n+1 \choose 2}}$ is notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".

The fact that the ${\displaystyle n}$th triangular number equals ${\displaystyle n(n+1)/2}$ can be illustrated using a visual proof. For every triangular number ${\displaystyle T_{n}}$, imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions ${\displaystyle n\times (n+1)}$, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: ${\displaystyle T_{n}={\frac {n(n+1)}{2}}}$. The example ${\displaystyle T_{4}}$ follows:

This formula can be proven formally using mathematical induction. It is clearly true for ${\displaystyle 1}$:

$${\displaystyle T_{1}=\sum _{k=1}^{1}k={\frac {1(1+1)}{2}}={\frac {2}{2}}=1.}$$

Now assume that, for some natural number ${\displaystyle m}$, ${\displaystyle T_{m}=\sum _{k=1}^{m}k={\frac {m(m+1)}{2}}}$. We can then verify it for ${\displaystyle m+1}$: $${\displaystyle {\begin{aligned}\sum _{k=1}^{m+1}k&=\sum _{k=1}^{m}k+(m+1)\\&={\frac {m(m+1)}{2}}+m+1\\&={\frac {m^{2}+m}{2}}+{\frac {2m+2}{2}}\\&={\frac {m^{2}+3m+2}{2}}\\&={\frac {(m+1)(m+2)}{2}},\end{aligned}}}$$