Question

若(2-x)^100=a0+a1x+a2x^2+……+a100x^100 求:

Answer 1

1.

令x=1

a0+a1+a2+...+a100=(2-1)^100=1^100=1

2.

令x=0

a0=(2-0)^100=2^100

3.

x为负时,对于an,n为奇数时an0

令x=-1

a0-a1+a2-a3+...+a100=[2-(-1)]^100=3^100

|a0|+|a1|+|a2|+...+|a100|

=a0-a1+a2-a3+...+a100

=3^100

4.

a0-a1+a2-a3+...+a100 (1)

a0+a1+a2+...+a100=1 (2)

(1)+(2)

2(a0+a2+...+a100)=3^100 +1

a0+a2+...+a100=(3^100 +1)/2

5.

(2)-(1)

2(a1+a3+...+a99)=1-3^100

a1+a3+...+a99=(1-3^100)/2