AP EAMCET201823 Apr 2018Evening ShiftMathematicsComplex NumberActual
For n Z⁺ , (1+ +i )^n+(1+ -i )^n=
Options
- A2^ n+1 ^n ( 4 - 2 ) ( n 4 - 2 )
- B2^ n+1 ^n ( 4 - 2 ) ( n 4 - 2 )
- C2^ n+1 ^n ( 4 - 2 ) ( n 4 - n 2 )
- D2^ n+1 ^n ( 4 - 2 ) ( n 4 - n 2 )
Correct answer
C. 2^ n+1 ^n ( 4 - 2 ) ( n 4 - n 2 )
Step-by-step solution
aligned & (1+ +i )^n+(1+ -i )^n & = [1+ ( 2 - )+i ( 2 - ) ] & + [ [1+ ( 2 - )-i ( 2 - ) ]^n . & = [2 ^2 ( 4 - 2 )+2 i ( 4 - 2 ) ( 4 - 2 ) ]^n aligned aligned & + [2 ^2 ( 4 - 2 )-2 i ( 4 - 2 ) ( 4 - 2 ) ]^n = & .2^n ^n ( 4 - 2 ) ) [ [ ( 4 - 2 )+i ( 4 - 2 ) ]^n . & .+ [ ( 4 - 2 )-i ( 4 - 2 ) ] = & 2^n ^n ( 4 - 2 ) [ ( n 4 - n 2 )+i ( n 4 - n 2 ) . & .+ ( n 4 - n 2 )-i ( n 4 - n 2 ) ] = & 2^ n+1 ^n ( 4 - 2 ) ( n 4 - n 2 ) . aligned