2024 AMC 12A Problem 21

Suppose that a_1 = 2 and the sequence (a_n) satisfies the recurrence relation \frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}for all n \ge 2. What is the greatest integer less than or equal to \sum^{100}_{n=1} a_n^2? \textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \qquad \textbf{(D) } 338{,}553 \qquad \textbf{(E) } 338{,}554

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.