Question

The nth triangular number is defined to be the sum of the first $n$ positive integers. For example, the $4^{th}$ triangular number is $1+2+3+4=10$. In the first $100$ terms of the sequence $1,3,6,10,15,21,28$ of triangular numbers, how many are divisible by $7$?

Answer 1

Given sequence: $1,3,6,10,15,21,..$

As we can observe that every $6^{th}$ and $7^{th}$ terms are divisible by $7.$

Therefore, total number of triangular numbers divisible by $7$ in first $100$ terms$=16+14-2=28$

(since, ${42}^{th}$ and ${84}^{th}$ terms are multiple of both)