Let the increasing sequence of integers D: d

_{1}, d

_{2}, d

_{3}, ...d

_{n-1}, d

_{n}...
represent the sequence of consecutive differences of another sequence T: t

_{1}, t

_{2}, t

_{3}, ...t

_{n-1}, t

_{n}.

Given t_{1}=0 , and t_{k}-t_{k-1}=d_{k-1}=k, provide definition and formula for the sequence T.

Steve H. identified the name of the sequence as "The Triangular Numbers". I would hate that fact to be lost. While our answers differed simply due to indices, he actually answered the challenge, which was to provide the name. I always learn something new here. I had to go to wikipedia to find the name origin. Interesting. So, I wondered if there are Tetrahedral Numbers, and indeed there are. They even count the total gifts in each verse of "The Twelve Days of Christmas" Thanks!

*Edited on ***September 17, 2018, 3:28 pm**