已知（x+2）9=a0+a1x+a2x2+……+a9x9,则（a1+3a3+5a5+7a7+9a9)2-（2a2+4a4+6a6+8a8)2的值是?

Question

已知（x+2）9=a0+a1x+a2x2+……+a9x9,则（a1+3a3+5a5+7a7+9a9)2-（2a2+4a4+6a6+8a8)2的值是?

Answer 1

(x+2)^9=a0+a1x+a2x^2+…+a9x^9
两边求导得
9(x+2)^8=a1+2a2x+3a3x^2…+9a9x^8
令x=1,
9(1+2)^8=a1+2a2+3a3…+9a9
9*3^8=a1+2a2+3a3…+9a9
3^2*3^8=a1+2a2+3a3…+9a9
3^10=a1+2a2+3a3…+9a9
a1+2a2+3a3…+9a9=3^10
令x=-1,
9(x+2)^8=a1+2a2x+3a3x^2…+9a9x^8
9(-1+2)^8=a1-2a2+3a3…+9a9
9*1^8=a1-2a2+3a3-4a4…+9a9
9=a1-2a2+3a3-4a4…+9a9
a1-2a2+3a3-4a4…+9a9=9
(a1+3a3+5a5+7a7+9a9)^2-(2a2+4a4+6a6+8a 8)^2
=[(a1+3a3+5a5+7a7+9a9)-(2a2+4a4+6a6+8a 8)][(a1+3a3+5a5+7a7+9a9)+(2a2+4a4+6a6+8a 8)]
=(a1+2a2+3a3…+9a9)(a1-2a2+3a3-4a4…+9a9)
=3^10*9
=3^10*3^2
=3^12