The famous Greek mathematician Pythagoras referred to the numbers obtained by the sum of the first positive integers as triangular numbers. For example, 1, 3, 6, and 10 are triangular numbers.

$$
\begin{aligned}
1 & =1 \\
3 & =1+2 \\
6 & =1+2+3 \\
10 & =1+2+3+4
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-006.jpg?height=300&width=717&top_left_y=575&top_left_x=1069)

The figure illustrates the motivation for the name of the triangular numbers. The sequence of triangular numbers continues with $1+2+3+4+5=15,1+2+3+4+5+6=21$ etc. How many triangular numbers are there less than 100?
We notice that the second triangular number is obtained from the first by adding 2, the third is obtained from the second by adding 3, and so on. This observation shows us how to calculate the next triangular numbers without doing many calculations. For example, we already know that the fourth triangular number is 10, so the fifth will be $10+5=15$ and the sixth will then be $15+6=21$. Thus, we can write the triangular numbers until we exceed 100.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-041.jpg?height=240&width=1034&top_left_y=1950&top_left_x=545)

Therefore, the triangular numbers less than 100 are $1, 3, 6, 10, 15, 21, 28, 36, 45$, 55, 66, 78, and 91. Thus, there are 13 triangular numbers less than 100.