Let `2(1+x^(3))^(100)=sum_(r=0)^(300) {a_(r)x^(r)-cos.(rpi)/(2)},"if"sum_(r=0)^(150) a_(2r)=s`, then `s-2^([log_(2)s]` is (where lfloor.rfloor denotes greatest integer function)

Answer

A

Explanation

`2(1+x^(3))^(100)=[a_(0)+a_(1)x+a_(2)x^(2)+....+a_(300)x^(300)]-1...(1)`
Replace x by -x, we get
`2(1-x^(3))^(100)=[a_(0)-a_(1)x+a_(2)x^(2)-...+a_(300^(x^(300)))]-1...(2)`
Additng (1)and (2),we get
`2(1+x^(3))^(100)+2(1-x^3)^(100)=2(a_(0)+a_(2)x^(2)+...+a_(300)x^(300))-2`
Put `x=1`,
we get `a_(0)+a_(2)+a_(4)+...+a_(300)=2^(100) + l = s`
`impliess-2^([log_(2)s])=2^(100)+1-2^(100)=1`