已知{an},{bn}都是各项为正数的数列,都有an,bn^2,an+1成等差数列 ;bn^2,an+1,bn+1^2成等比数列,若a1=1,b1=根号2,求sn=1/a1+1/a2+…1/an
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![已知{an},{bn}都是各项为正数的数列,都有an,bn^2,an+1成等差数列 ;bn^2,an+1,bn+1^2成等比数列,若a1=1,b1=根号2,求sn=1/a1+1/a2+…1/an](/uploads/image/z/4027197-21-7.jpg?t=%E5%B7%B2%E7%9F%A5%7Ban%7D%2C%7Bbn%7D%E9%83%BD%E6%98%AF%E5%90%84%E9%A1%B9%E4%B8%BA%E6%AD%A3%E6%95%B0%E7%9A%84%E6%95%B0%E5%88%97%2C%E9%83%BD%E6%9C%89an%2Cbn%5E2%2Can%2B1%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97+%EF%BC%9Bbn%5E2%2Can%2B1%2Cbn%2B1%5E2%E6%88%90%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E8%8B%A5a1%3D1%2Cb1%3D%E6%A0%B9%E5%8F%B72%2C%E6%B1%82sn%3D1%2Fa1%2B1%2Fa2%2B%E2%80%A61%2Fan)
已知{an},{bn}都是各项为正数的数列,都有an,bn^2,an+1成等差数列 ;bn^2,an+1,bn+1^2成等比数列,若a1=1,b1=根号2,求sn=1/a1+1/a2+…1/an
已知{an},{bn}都是各项为正数的数列,都有an,bn^2,an+1成等差数列 ;bn^2,an+1,bn+1^2成等比数列,若a1=1,b1=根号2,求sn=1/a1+1/a2+…1/an
已知{an},{bn}都是各项为正数的数列,都有an,bn^2,an+1成等差数列 ;bn^2,an+1,bn+1^2成等比数列,若a1=1,b1=根号2,求sn=1/a1+1/a2+…1/an
∵bn²,a(n+1),b(n+1)²成等比数列
∴bn² × b(n+1)²= a(n+1)²
∵bn>0,b(n+1)>0,a(n+1)>0
∴bn × b(n+1) = a(n+1)……①
∴b(n-1)×bn = an……②
∵an,bn²,a(n+1)成等差数列.
∴2bn²=an+a(n+1)……③
把①②带入得,2bn²=b(n-1)×bn + bn × b(n+1)
约去bn,得,2bn=b(n-1) + b(n+1),n≥2
∴数列{bn}是等差数列.
∵a1=1,b1=√2,由③得,a2=3
再由①,得b2=3/√2 = 3√2/2
∴公差d=b2-b1=√2/2
∴bn=b1+(n-1)d=√2+ (√2/2)(n-1)=(√2/2)(n+1)
∴b(n-1)=(√2/2)n
由②,得:an=n(n+1)/2
∴sn=1/a1+1/a2+……1/an
=2/(1×2) + 2/(2×3) + …… + 2/[n×(n+1)]
=2[1-(1/2)+(1/2)-(1/3)+……+(1/n)-1/(n+1) ]
=2[1 - 1/(n+1)]
=2n/(n+1)