A) \[{{a}^{2}}+{{b}^{2}}\]
B) \[\sqrt{{{a}^{2}}+{{b}^{2}}}\]
C) \[\sqrt{{{a}^{2}}-{{b}^{2}}}\]
D) \[{{a}^{2}}-{{b}^{2}}\]
E) \[a+b\]
Correct Answer: A
Solution :
\[\because \]\[(1+i)(1+2i)(1+3i)....(1+{{n}_{i}})=a+ib\] ...(i) \[\Rightarrow \]\[(1-i)(1-2i)(1-3i)....(1-ni)=a-ib\] ...(ii) \[\therefore \]From Eqs. (i) and (ii), we get \[(1-{{i}^{2}})(1-4{{i}^{2}})(1-9{{i}^{2}})....(1-{{n}^{2}}{{i}^{2}})\] \[={{a}^{2}}+{{b}^{2}}\] \[\Rightarrow \]\[(1+1)(1+4)(1+9)....(1+{{n}^{2}})={{a}^{2}}+{{b}^{2}}\] \[\Rightarrow \] \[2.5.10......(1+{{n}^{2}})={{a}^{2}}+{{b}^{2}}\]
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