Question

设(x+2)^9=a0+a1x+a2x^2+...+a9x^9 求（a1+3a3+5a5+7a7+9a9）^2--（2a2+4a4+6a6+8a 8)^2

Answer 1

解：当x＝1时，a0＋a1＋a2＋a3＋a4＋a5＋a6＋a7＋a8＋a9＝19683 ①
当x＝－1时，a0－a1＋a2－a3＋a4－a5＋a6－a7＋a8－a9＝1 ②
当x＝0时， a0＝2^9＝512
①－②得：a1＋a3＋a5＋a7＋a9＝9841
①＋②得：a0＋a2＋a4＋a6＋a8＝9842
a2＋a4＋a6＋a8＝9330
（a1＋a3＋a5＋a7＋a9）²－（a2＋a4＋a6＋a8）²＝9841²－9330²
＝（9841－9330）×（9841＋9330）
＝9796381

Answer 2

(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
这个对

Answer 3

3^12

Answer 4

(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