`("^(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=` (a) 4k (b) 4k+1 (c) 4k-1 (d) 4k+2

Options

A. `4k`
B. `4k+1`
C. `4k-1`
D. `4k+2`

Answer

C

Explanation

`(c )` Consider,
`(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`