InterviewSolution
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("^(m)C_(0)+^(m)C_(1)-^(m)C_(2)-^(m)C_(3))+('^(m)C_(4)+^(m)C_(5)-^(m)C_(6)-^(m)C_(7))+..=0 if and only if for some positive integer k, m= |
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Answer» `4k` `(costheta-isintheta)^(m)` `=.^(m)C_(0)cos^(m)theta-^(m)C_(1)cos^(m-1)thetaisintheta+...+^(m)C_(m)(-isintheta)^(m)`.......`(i)` `(costheta+isintheta)^(m)` `=^(m)C_(0)cos^(m)theta+^(m)C_(1)cos^(m-1)thetaisintheta+...+^(m)C_(m)(isintheta)^(m)`.......`(II)` Adding `(1)` and `(2)` , we GET `2cosmtheta=2['^(m)C_(0)cos^(m)theta-^(m)C_(2)cos^(m-2)thetasin^(2)theta....]`.......`(iii)` Subtracting `(1)` from `(2)`, we get `2isinmtheta=2i['^(m)C_(1)cos^(m-1)thetasintheta-^(m)C_(3)cos^(m-3)thetasin^(3)theta.....]`.....`(iv)` Adding `(3)` and `(4)`, we get `cosmtheta+sinmtheta` `=['^(m)C_0)cos^(m)theta+^(m)C_(1)cos^(m-1)thetasintheta-^(m)C_(2)cos^(m-2)thetasin^(2)theta-^(m)C_(3)cos^(m-3)thetasin^(3)theta....]` `impliessqrt(2)sin(mtheta+(pi)/(4))` `=['^(m)C_(0)cos^(m)theta+^(m)C_(1)cos^(m-1)thetasintheta-^(m)C_(2)cos^(m-2)thetasin^(2)theta-^(m)C_(3)cos^(m-3)thetasin^(3)theta...]` Putting `theta=(pi)/(4)`, we get `sqrt(2)sin"(((m+1)pi)/(4))` `=(1)/(2^(m//2))` `[('^(m)C_(0)+^(m)C_(1)-^(m)C_(2)-^(m)C_(3))+('^(m)C_(4)+^(m)C_(5)-^(m)C_(6)-^(m)C_(7))+...+('^(m)C_(m-3)+^(m)C_(m-2)-^(m)C_(m-1)-^(m)C_(m))]` Hence , `m+1=4k`, for given quantity to be `0`. `implies m=4k-1`, where `K in N` |
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