jmc

The real numbers in the sum follow the pattern $$-2+4-6+8-10+12+\cdots .$$Placing the numbers in pairs, we get $$(-2+4)+(-6+8)+(-10+12)+\cdots =2+2+2+\cdots .$$Thus, we want 24 pairs. The second number in the $n$th pair is $4n,$ which means the last real term is $96=96i^{96}.$ Thus, $n$ is either 96 or 97.

Up to $96i^{96},$ the sum of the imaginary numbers is $$i-3i+5i-7i+9i-11i+\cdots +93i-95i.$$Similarly, if we pair these numbers, we get $$(i-3i)+(5i-7i)+(9i-11i)+\cdots +(93i-95i)=-2i-2i-2i-\cdots -2i=-48i.$$Adding $97i^{97}=97i,$ we get $97i-48i=49i.$ Hence, $n=\boxed{97}.$