If x2k occurs in the expansion of x+1x2n−3, then

Tr+1 the (r+1)th term in the expansion of x+1x2n−3is given byTr+1=n−3Cr(x)n−3−r1x2r=n−3Crxn−3−3rAs x2k occurs in the expansion of x+1x2n−3,we must haven n-3-3r=2k for some non-negative integer r.⇒  3(1+r)=n-2k    ⇒ n-2k is a multiple of 3.