A) \[50(1-i)\]
B) \[25i\]
C) \[.25\,(1+i)\]
D) \[100(1-i)\]
Correct Answer: A
Solution :
[a] Let \[S=i-2-3i+4+5i....100\] terms \[\Rightarrow S=i+2{{i}^{2}}+3{{i}^{3}}+4{{i}^{4}}+5{{i}^{5}}....+100{{i}^{100}}\] \[\Rightarrow iS={{i}^{2}}+2{{i}^{3}}+3{{i}^{4}}+99{{i}^{100}}+100{{i}^{101}}\] \[\Rightarrow S-iS=i+{{i}^{2}}+{{i}^{3}}+{{i}^{4}}+....+{{i}^{100}}-100{{i}^{101}}\] \[\Rightarrow S(1-i)=\frac{i(1-{{i}^{100}})}{1-i}-100{{i}^{101}}\] \[\Rightarrow S(1-i)=-100i\] \[\Rightarrow S=\frac{-100i}{1-i}=-50i(1+i)=-50(i-1)\] \[=50(1-i)\]
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