Question

已知:x=[(√5)+1]/2,求(x^3+5x+1)/x^5的值~

已知:x=[(√5)+1]/2,求(x^3+5x+1)/x^5的值~
能做的让人看的懂么...

Answer 1

已知:x=[(√5)+1]/2,求(x^3+5x+1)/x^5的值。

解：由已知变形，得
2x=√5+1
2x-1=√5
两边同时平方，得
4x^2-4x+1=5
x^2-x-1=0
x^2=x+1
上式两边同乘以x，得：
x^3=x^2+x=(x+1)+x=2x+1············注：将x^2=x+1代入；
上式两边又再同时乘以x^2，得
x^5=2x^3+x^2=2(2x+1)+(x+1)=5x+3·····注：将x^3=2x+1和x^2=x+1分别代入；
所以
(x^3+5x+1)/x^5
=(2x+1+5x+1)/(5x+3)·······注：将x^3=2x+1和x^5=5x+3分别代入；
=(7x+2)/(5x+3)
=(14x+4)/(10x+6)······注：为了方便，分子分母同乘以2；
=[7(2x-1)+11]/[5(2x-1)+11]······注：将2x-1=√5代入；
=(7√5+11)/(5√5+11)
=(7√5+11)(5√5-11)/[(5√5+11)(5√5-11)]
=(54-22√5)/(125-121)
=(27-11√5)/2

为了让您看更清楚一些，写得比较详细了点，实际解题时部分过程可以省略。

Answer 2

x=(5^0.5+1)/2=2/(5^0.5-1)
1/x=(5^0.5-1)/2

(x^3+5x+1)/x^5=1/x^2+5/x^4+1/x^5=[(5^0.5-1)/2]^2+5*[(5^0.5-1)/2]^4+[(5^0.5-1)/2]^5

[(5^0.5-1)/2]^2=(3-5^0.5)/2
[(5^0.5-1)/2]^4=(7-3*5^0.5)/2
[(5^0.5-1)/2]^5=(5*5^0.5-11)/2

(x^3+5x+1)/x^5=(27-11*5^0.5)/2