Sum of Triangular Numbers

triangular numbers are those numbers of objects that could create an equilateral triangle. like 3, 6, 10, etc.. Between 1 and 100, these are the triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78 and 91 which sum up to 455.

Now, we know that a triangular number can be represented by the formula :- $\dfrac {(n)(n+1)}{2}$
When $n =1$ triangular number = $1$
When $n =2$ triangular number = $3$
When $n =3$ triangular number = $6$
When $n =4$ triangular number = $10$
When $n =5$ triangular number = $15$
When $n =6$ triangular number = $21$
When $n =7$ triangular number = $28$
When $n =8$ triangular number = $36$
When $n =9$ triangular number = $45$
When $n =10$ triangular number = $55$
When $n =11$ triangular number = $66$
When $n =12$ triangular number = $78$
When $n =13$ triangular number = $91$
When $n =14$ triangular number = $105$

So, we see that the largest triangular number smaller than $100$ is $91$ for which $(n = 13)$

So, all we have to do is evaluate $\displaystyle \sum_{n=1}^{13}$ = $\boxed {455}$ .

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91