The sum of $$i - 2 - 3i + 4....\,{\text{upto }}100\,{\text{terms,}}$$       where $$i = \sqrt { - 1} $$   is

Solution :
Let $$S = i - 2 - 3i + 4 + 5i.... 100\,{\text{terms,}}$$
$$\eqalign{ & \Rightarrow S = i + 2{i^2} + 3{i^3} + 4{i^4} + 5{i^5}.... + 100{i^{100}} \cr & \Rightarrow iS = {i^2} + 2{i^3} + 3{i^4}.... + 99{i^{100}} + 100{i^{101}} \cr & \Rightarrow S - iS = i + {i^2} + {i^3} + {i^4} + .... + {i^{100}} - 100{i^{101}} \cr & \Rightarrow S\left( {1 - i} \right) = \frac{{i\left( {1 - {i^{100}}} \right)}}{{1 - i}} - 100{i^{101}} \cr & \Rightarrow S\left( {1 - i} \right) = - 100i \cr & \Rightarrow S = \frac{{ - 100i}}{{1 - i}} = - 50i\left( {1 + i} \right) = - 50\left( {i - 1} \right) \cr & = 50\left( {1 - i} \right) \cr} $$