Sure! The $n$th triangle number is given by the formula:
$T(n) = (n) + (n-1) + (n-2) + ... + 1$
The name "triangle number" comes from the fact that if you build a triangle of height $n$, the number of elements in the triangle will be the $n$th triangle number. Here are the first few examples:
```
 * (n=2, T(n) = 3)
**
```

```
   * (n=3, T(n)=6
  **
 ***
```

If we continue this pattern we will see that the base of the triangle is length $n$ and the height of the triangle is length $n$ as well. The formula for the area of the triangle is $(b \times h) / 2$ or in this case $\frac{n^2}{2}$. That is close to the formula for the $n$th triangle number which is $\frac{n(n+1)}{2}$. The reason we increase the second $n$ by 1 is because we aren't getting the real area, but rather counting dots.
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence