等比数列{an}的各项均为正数,且2a1+3a2=1,a3²=9a2a6,求通项公式
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![等比数列{an}的各项均为正数,且2a1+3a2=1,a3²=9a2a6,求通项公式](/uploads/image/z/5293240-16-0.jpg?t=%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E7%9A%84%E5%90%84%E9%A1%B9%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B0%2C%E4%B8%942a1%2B3a2%3D1%2Ca3%26%23178%3B%3D9a2a6%2C%E6%B1%82%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F)
等比数列{an}的各项均为正数,且2a1+3a2=1,a3²=9a2a6,求通项公式
等比数列{an}的各项均为正数,且2a1+3a2=1,a3²=9a2a6,求通项公式
等比数列{an}的各项均为正数,且2a1+3a2=1,a3²=9a2a6,求通项公式
a3²=9a2a6
9q²=1
q=1/3
2a1+3a2=1
2a1+3a1*1/3=1
a1=1/3
an=(1/3)^n
(1)a3^2=9a2a6
(a2p)^2=9a2(a2p^4)
a2^2p^2=9a2^2p^4
∵此数列各项均为正数∴a2^2<>0,p>0
两边同时除以a2^2p^2,得9p^2=1,p=1/3
2a1+3a2=1
2a1+3*[(1/3)*a1]=1
2a1+a1=1
3a1=1
a1=1/3
an=a1p^(n-1)=1/3*(1/3)^(n-1)=1/3^n
(1)a3^2=9a2a6
(a2p)^2=9a2(a2p^4)
a2^2p^2=9a2^2p^4
∵此数列各项均为正数∴a2^2<>0,p>0
两边同时除以a2^2p^2,得9p^2=1,p=1/3
2a1+3a2=1
2a1+3*[(1/3)*a1]=1
2a1+a1=1
3a1=1
a1=1/3
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(1)a3^2=9a2a6
(a2p)^2=9a2(a2p^4)
a2^2p^2=9a2^2p^4
∵此数列各项均为正数∴a2^2<>0,p>0
两边同时除以a2^2p^2,得9p^2=1,p=1/3
2a1+3a2=1
2a1+3*[(1/3)*a1]=1
2a1+a1=1
3a1=1
a1=1/3
an=a1p^(n-1)=1/3*(1/3)^(n-1)=1/3^n
(2)bn=log3a1+log3a2+...+log3an
=log3(a1*a2*...*an)
=log3[(1/3)*(1/3^2)*...*(1/3^n)]
=log3[(1/3)^(1+2+...+n)]
=(1+2+...+n)*log3(1/3)
=-n(n+1)/2
1/bn=-2/n(n+1)
=(-2)*[1/n(n+1)]
=(-2)*[1/n-1/(n+1)]
1/b1+1/b2+...+1/bn
=(-2)*(1-1/2)+(-2)*(1/2-1/3)+...+(-2)*[1/n-1/(n+1)]
=(-2)*[1-1/2+1/2-1/3+...+1/n-1/(n+1)]
=(-2)*[1-1/(n+1)]
=-2n/(n+1)
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