1道二项式定理题设（2-根号3乘x）^100=a0+a1x+a2x²+...a下标100x^100,则（a0+a2+a4+...a100)²-(a1+a3+a5+...a99)²的值是

Question

1道二项式定理题
设（2-根号3乘x）^100=a0+a1x+a2x²+...a下标100x^100,则（a0+a2+a4+...a100)²-(a1+a3+a5+...a99)²的值是

Answer 1

设f(x)=(2-√3乘x）^100
（a0+a2+a4+...a100)²-(a1+a3+a5+...a99)²
=(a0+a1+a2+.+a100)(a0-a1+a2-a3+a4-a5+.a99-a100)
=f(1)*f(-1)
=(2-√3）^100*(2+√3）^100
=[(2-√3)(2+√3)]^100
=1^100
=1