数列{an}满足:a1=2,an=[4a(n-1)]^5,求{an}的通项

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数列{an}满足:a1=2,an=[4a(n-1)]^5,求{an}的通项

数列{an}满足:a1=2,an=[4a(n-1)]^5,求{an}的通项
数列{an}满足:a1=2,an=[4a(n-1)]^5,求{an}的通项

数列{an}满足:a1=2,an=[4a(n-1)]^5,求{an}的通项
a(n+1)=[4a(n)]^5,
a(1)=2>0,a(n)>0.
ln[a(n+1)]=5ln[4a(n)] = 5ln[a(n)] + 5ln(4),
ln[a(n+1)] + x = 5ln[a(n)] + 5ln(4) + x = 5{ln[a(n) + ln(4) + x/5} = 5{ln[a(n)] + x},
ln(4) + x/5 = x,x = (5/4)ln(4),
ln[a(n+1)] + (5/4)ln(4) = 5ln[a(n)] + 5ln(4) + (5/4)ln(4) = 5{ln[a(n)] + (5/4)ln(4)},
{ln[a(n)] + (5/4)ln(4)}是首项为ln[a(1)] + (5/4)ln(4)=ln(2) + (5/4)ln(4)=ln(2)+(5/2)ln(2)=(7/2)ln(2),公比为5的等比数列.
ln[a(n)] + (5/4)ln(4) = (7/2)ln(2)*5^(n-1),
ln[a(n)] = (7/2)ln(2)*5^(n-1) - (5/2)ln(2) = [(7/2)5^(n-1) - 5/2]ln(2) = ln{2^[(7/2)5^(n-1) - 5/2]},
a(n) = 2^[(7/2)5^(n-1) - 5/2]