F.2 Maths

As follows~

圖片參考：http://i204.photobucket.com/albums/bb125/jasonchan424/746.jpg

anw=1

(z + 1/z)^2 = 9
z^2 + 2 + 1/z^2 = 9
z^2 + 1/z^2 = 7
(z^2 + 1/z^2)^2 = 49
z^4 + 2 + 1/z^4 = 49
z^4 + 1/z^4 = 47
(z + 1/z)^6
= z^6 + 6(z^5)(1/z) + 15(z^4)(1/z^2) + 20(z^3)(1/z^3) + 15(z^2)(1/z^4)
+ 6(z)(1/z^5) + 1/z^6
= z^6 + 6z^4 + 15z^2 + 20 + 15/z^2 + 6/z^4 + 1/z^6
= z^6 + 1/z^6 + 6z^4 + 6/z^4 + 15z^2 + 15/z^2 + 20
= (z^6 + 1/z^6) + 6(z^4 + 1/z^4) + 15(z^2 + 1/z^2) + 20
= (z^6 + 1/z^6) + 6(47) + 15(7) + 20
= (z^6 + 1/z^6) + 407
(z + 1/z)^6 = (z^6 + 1/z^6) + 407
3^6 = (z^6 + 1/z^6) + 407
z^6 + 1/z^6 = 322

2007-10-13 20:51:06 補充：
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z+1/z=3

square both sides,
(z+1/z)^2 =9
z^2 + 2 (z)(1/z) +1/z^2 =9
z^2+1/z^2 = 7

Then
(z^2+1/z^2)^3 = 7^3
(z^2)^3 + 3(z^2)^2 (1/z^2) + 3(z^2)(1/z^2)^2 +(1/z^2)^3 = 343
z^6 + 3z^2 + 3/z^2 + 1/z^6 = 343
z^6+1/z^6 + 3(z^2+1/z^2) = 343
z^6+1/z^6 = 343 -3(7)
= 322

z^6+1/z^6
=(z^2)^3+(1/z^2)^3
=(z^2+1/z^2)^3 - 3(1/z^2)(z^2)^2-3(z^2)(1/z^2)^2
=((z+1/z)^2-2)^3-3(z^2)-3(1/z^2)
=(3^2-2)^3-3(z^2+1/z^2)
=7^3-3*7
=343-21
=322

As follows~~~


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