已知a_1=4,a_(n+1)=(〖a_n〗^2+4)/(2a_n ),求数列通项公式

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已知a_1=4,a_(n+1)=(〖a_n〗^2+4)/(2a_n ),求数列通项公式

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a(n+1)=(an²+4)/(2an)
a(n+1) +2=(an²+4an+4)/(2an)=(an +2)²/(2an)
a(n+1)-2=(an²-4an+4)/(2an)=(an -2)²/(2an)
[a(n+1)+2]/[a(n+1)-2]=[(an +2)/(an -2)]²
n=...

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a(n+1)=(an²+4)/(2an)
a(n+1) +2=(an²+4an+4)/(2an)=(an +2)²/(2an)
a(n+1)-2=(an²-4an+4)/(2an)=(an -2)²/(2an)
[a(n+1)+2]/[a(n+1)-2]=[(an +2)/(an -2)]²
n=1时,(a1+2)/(a1-2)=(4+2)/(4-2)=3>0,假设当n=k(k∈N+)时,(ak+2)/(ak-2)>0
则当n=k+1时,
[a(k+1)+2]/[a(k+1)-2]=[(ak+2)/(ak-2)]²>0
k为任意正整数,因此对于任意正整数n,数列{(an +2)/(an -2)}恒>0
log3[(a(n+1)+2)/(a(n+1)-2)]=log3[(an +2)/(an -2)]²=2log3[(an +2)/(an -2)]
log3[(a(n+1)+2)/(a(n+1)-2)]/log3[(an +2)/(an -2)]=2,为定值
log3[(a1+2)/(a1-2)]=log3(3)=1,数列{log3[(an +2)/(an -2)]}是以1为首项,2为公比的等比数列
log3[(an +2)/(an -2)]=2^(n-1)
(an +2)/(an -2)=3^[2^(n-1)]
{3^[2^(n-1)]-1}an=2×3^[2^(n-1)]+2
an={2×3^[2^(n-1)]+2}/{3^[2^(n-1) -1]}

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