已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn (1)计算:S1=,S2=,S3=已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn(1)计算:S1=,S2=,S3=,S4=;(2)试写出Sn-2、Sn-1、Sn三者之间
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已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn (1)计算:S1=,S2=,S3=已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn(1)计算:S1=,S2=,S3=,S4=;(2)试写出Sn-2、Sn-1、Sn三者之间
已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn (1)计算:S1=,S2=,S3=
已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn
(1)计算:S1=,S2=,S3=,S4=;
(2)试写出Sn-2、Sn-1、Sn三者之间的关系;
(3)根据以上得出结论计算:α7+β7.
已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn (1)计算:S1=,S2=,S3=已知α+β=1,αβ=-1.设S1=α+β,S2=α2+β2,S3=α3+β3,…,Sn=αn+βn(1)计算:S1=,S2=,S3=,S4=;(2)试写出Sn-2、Sn-1、Sn三者之间
(1)∵α+β=1,αβ=-1.
∴S1=α+β=1.
S2=α^2+β^2=(α+β)^2-2αβ=1+2=3.
S3=α^3+β^3=(α+β)(α^2-αβ+β^2)=(α+β)^2-3αβ=1+3=4.
S4=α^4+β^4=(α^2+β^2)^2-2α^2β^2=9-2=7.
故答案为:1,3,4,7;
(2)由(1)得:Sn=Sn-1+Sn-2.
证明:∵α,β是方程x^2-x-1=0的两根,
∴有:α^2=α+1,β^2=β+1,
Sn-1+Sn-2=α^n-1+β^n-1+α^n-2+β^n-2
=αn/α +αn/α2 +βn/β+βn/β2 =α^n(1+α)/α^2 +β^n(1+β)/β^2
=α^n+β^n
=Sn.
故Sn=Sn-1+Sn-2.
(3)由(2)有:
α^7+β^7=S7
=S6+S5
=S5+S4+S4+S3
=S4+S3+2S4+S3
=3S4+2S3
=3×7+2×4
=29.