AMC 12 2024a 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 $$\sum^{100}_{n=1} a_n^2?$$ $\textbf{(A) } 338{,}550 \textbf{(B) } 338{,}551 \textbf{(C) } 338{,}552 \textbf{(D) } 338{,}553 \textbf{(E) } 338{,}554$