2009 AMC 12A Problem 15

For $k$ a multiple of $4,$ $$\begin{aligned} &(k + 1)i^{k+1} + (k + 2)i^{k+2} \\ &\quad {}+ (k + 3)i^{k+3} + (k + 4)i^{k+4} \\ &= (k + 1)i - (k + 2) \\ &\quad {}- (k + 3)i + (k + 4) \\ &= 2 - 2i. \end{aligned}$$ Summing the first $96$ terms (that is $24$ blocks) gives $24(2 - 2i) = 48 - 48i.$ Adding the next term $97i^{97} = 97i$ yields $48 - 48i + 97i = 48 + 49i.$ So $n = 97.$ Thus, the correct answer is D.