`("^(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

If `theta in R and i = sqrt(-1), "then " (cos theta + i sin theta )^(n)`
`=^(m)C_(0) (cos theta)^(m) +^(m) C_(1)(costheta)^(m-1) (isin theta)`
`+ C_(2)(costheta)^(m-1) (isin theta)^(2)+...+""^(m)C_(m)(isintheta)^(m) `
`(cos m theta + isin m theta) = [""^(m)C_(0)(cos theta)^(m)-^(m)C_(2)(cos theta)^(m-2)cdot sin ^(2) theta +""^(m)C_(4)(costheta)^(m-4)sin^(4) theta - ...] + i [""^(m)C_(1)(cos theta)^(m-1) sin theta -^(m) C_(3) (cos theta)^(m-3)sin^(3) theta +..]`
[ using Demovire' s theromem]
Comparing real and imaginary parts, we geta `cos mtheta = ^(m)(cos theta)^(m) -^(m) C_(2) (cos theta)^(m-2)sin^(2)theta+^(m)C_(4) (cos theta )^(m-4) sin ^(4) theta - ......(i)`
`sin mtheta= ^(m) C_(1) (cos theta)^(m-1) cdot sintheta -^(m) C_(3)(cos theta)^(m-3 )cdot sin^(3) theta + ......(ii) `
On adding Eqs. (i) and (ii), we get
`cos m theta + sin m theta = ^(m)C_(0)(costheta)^(m) +
C_(1) (costheta)^(m-1)cdot sin theta -^(m) C_(2) (cos theta)^(m-2)sin ^(2) theta-^(m) C_(3)(cos)^(m-3)sin ^(3) theta`
`+^(m)C_(4)(costheta)^(m-4)sin^(4)theta + ...sin (mtheta + pi/4)`
`=(costheta)^(m){{:(""^(m)C_(0)+^(m)C_(1)tan theta-^(m)C_(2)tan^(2)theta -^(m)C_(3)tan^(3)),(+^(m)C_(4)tan^(4)theta + ^(m) C_(5)tan^(5) theta-...):}} `
Putting `theta=pi/4, 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)) ):}}`
`because((""^(m)C_(0) +^(m) C_(1)-^(m) C_(2)-^(m) C_(3))+(""^(m)C_(4)+^(m) C_(5)-^(m) C_(6)-^(m) C_(7))`
`therefore sin frac((m+1)pi)(4)=0 rArr ((m+1)pi)/4 = k pi`
or `m=4k-1,AAkinI`