Arithmetic Series versus Arithmetic Progression AIME 1989/7 says
If the integer k is added to each of the numbers 36, 300, and 596, one obtains the squares of three consecutive terms of an arithmetic series. Find k.
On looking at the solution, it becomes clear that the three terms considered are those of an arithmetic progression:
a, a+d, and a+2d
Shouldn't an arithmetic series be the sum of terms of an arithmetic progression, making it
a, 2a + d, 3a + 3d
 A: The original AIME version of this question, which can be found at https://gogangsa.com/338 , says arithmetic sequence, not arithmetic series. It's not clear how or why the AoPS version changed it, but people do slip up from time to time. The OP is quite correct to question the interpretation of the wording in the version they came across.
A: $(a+2d)$ commonly refers to the third term of both an arithmetic sequence/progression and an arithmetic series (e.g., here and here), so the cited source is correct.
On the other hand, the sum $(3a+3d)$ of their first three terms is technically called the third partial sum of the series (as opposed to the infinite series, i.e., the sum of the infinitely many terms of the series).
So $(a,\, 2a + d,\, 3a + 3d,\, ...)$ is actually a sequence of partial sums—not an arithmetic series as you had thought. To reiterate: the $n^{th}$ term of this sequence of partial sums is not the $n^{th}$ term of the series, but rather the series (sum) up the the $n^{th}$ term.
