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已知x2分1次方+x-2分1次方=根号3,当n属于N*时,记Mn=xn次方+x-n次方.
(1)求M1 M2 M3的值
(2)求证M1Mn+1=Mn+2+Mn
(3)求M2012的值
50分
人气:155 ℃ 时间:2020-04-05 03:30:24
解答
(1)∵x2分1次方+x-2分1次方=根号3
两边平方得x +x^-1 =1,即M1=1
再两边平方得x^2 +x^-2 =-1,M2=-1
而x^3 +x^-3=(x +x^-1)^3-3(x *x^-1)(x +x^-1)=-2,M3=-2
(2)M1Mn+1=(x +x^-1)(x^(n+1)+x^-(n+1))
=(x^(n+2)+x^-(n+2))+(x^n +x^-n)= Mn+2+Mn
(3)M2012=-2011
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