已知数列满足:a1=1,a(n+1)=an/(an+2),若b(n+1)=(n-a)(1/an+1),b1=-a,且数列{bn}是单调递增数列求实数a的取值范围
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![已知数列满足:a1=1,a(n+1)=an/(an+2),若b(n+1)=(n-a)(1/an+1),b1=-a,且数列{bn}是单调递增数列求实数a的取值范围](/uploads/image/z/13132846-46-6.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%E6%BB%A1%E8%B6%B3%3Aa1%3D1%2Ca%28n%2B1%29%3Dan%2F%28an%2B2%29%2C%E8%8B%A5b%28n%2B1%29%3D%28n-a%29%281%2Fan%2B1%29%2Cb1%3D-a%2C%E4%B8%94%E6%95%B0%E5%88%97%EF%BD%9Bbn%EF%BD%9D%E6%98%AF%E5%8D%95%E8%B0%83%E9%80%92%E5%A2%9E%E6%95%B0%E5%88%97%E6%B1%82%E5%AE%9E%E6%95%B0a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
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已知数列满足:a1=1,a(n+1)=an/(an+2),若b(n+1)=(n-a)(1/an+1),b1=-a,且数列{bn}是单调递增数列求实数a的取值范围
已知数列满足:a1=1,a(n+1)=an/(an+2),若b(n+1)=(n-a)(1/an+1),b1=-a,且数列{bn}是单调递增数列
求实数a的取值范围
已知数列满足:a1=1,a(n+1)=an/(an+2),若b(n+1)=(n-a)(1/an+1),b1=-a,且数列{bn}是单调递增数列求实数a的取值范围
a(n+1)=an/(an+2)
1/a(n+1)=(an+2)/an=2/an +1
1/a(n+1) +1=2/an +2=2(1/an +1)
[1/a(n+1)+1]/(1/an +1)=2,为定值.
1/a1 +1=1/1+1=2
数列{1/an +1}是以2为首项,2为公比的等比数列.
1/an +1=2ⁿ
b(n+1)=(n-a)/(1/an +1)=(n-a)/2ⁿ
b2=(1-a)/2
b2>b1 (1-a)/2>-a
a>-1
n≥1时,b(n+2)>b(n+1)
[(n+1)-a]/2^(n+1) -(n-a)/2ⁿ>0
(n+1)-a -2(n-a)>0
a>n-1
n≥1 n-1≥0 a≥0
综上,得a≥0
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