9 (x+2) 8 = a1+2a2x+3a3x2 ~ ~ 9a9x8 is used for simultaneous derivation on both sides.

Let x= 1 and get a1+2a2+3a3+~+9a9 = 9× (1+2) 8 = 310.

Let x=- 1 get a1-2a2+3a3 ~-8a8+9a9 = 9× (2-1) 8 = 9.

∴(a 1+3a3+5a5+7a7+9a9)^2-(2a2+4a4+6a6+8a8)^2

=(a 1+2 a2+3 a3+~ ~+9 a9)(a 1-2 a2+3 a3 ~ ~-8 A8+9 a9)

=9×3^ 10

=3^ 12