设数列{an}满足a1=1, an=(4an-1 +2)/(2an-1 +7) ,则通项xn=?

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an=[4a(n-1)+2]/[2a(n-1)+7]
an+2=[4a(n-1)+2+4a(n-1)+14]/[2a(n-1)+7]
=[8a(n-1)+16]/[2a(n-1)+7]
=8[a(n-1)+2]/[2a(n-1)+7]
an -1/2=[4a(n-1)+2-a(n-1)-7/2]/[2a(n-1)+7]
=[3a(n-1)-3/2]/[2a(n-1)+7]
=3[a(n-1)-1/2]/[2a(n-1)+7]
[(an +2)/(an -1/2)]/{[a(n-1)+2]/[a(n-1)-1/2]}=8/3,为定值.
(a1+2)/(a1-1/2)=(1+2)/(1-1/2)=6
数列{(an +2)/(an -1/2)}是以6为首项,8/3为公比的等比数列.
(an +2)/(an -1/2)=6×(8/3)^(n-1)
[6×(8/3)^(n-1) -1]an=3×(8/3)^(n-1)+2
an=[3×(8/3)^(n-1) +2]/[6×(8/3)^(n-1) -1]
=[3×(8/3)^(n-1) -1/2 +5/2]/[6×(8/3)^(n-1) -1]
=(1/2) +5/[12×(8/3)^(n-1) -2]
=(1/2) +5/[2^(3n-1)/3^(n-2) -2]
n=1时,a1=(1/2) +5/[2²/3^(-1) -2]=(1/2)+5/10=1,同样满足.
数列{an的通项公式为an=(1/2) +5/[2^(3n-1)/3^(n-2) -2].

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