已知数列{an}的前n项和为Sn,且a1=1/2,an+1=(n+1)an/2n.(1)求数列{an}的通项公式;(2)设bn=n(2-Sn),n∈N*,若集合M={n|bn≥λ,n∈N*}恰有4个元素,求实数λ的取值范围.
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![已知数列{an}的前n项和为Sn,且a1=1/2,an+1=(n+1)an/2n.(1)求数列{an}的通项公式;(2)设bn=n(2-Sn),n∈N*,若集合M={n|bn≥λ,n∈N*}恰有4个元素,求实数λ的取值范围.](/uploads/image/z/400410-18-0.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E4%B8%94a1%3D1%2F2%2Can%2B1%3D%28n%2B1%29an%2F2n.%281%29%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B%EF%BC%882%EF%BC%89%E8%AE%BEbn%3Dn%282-Sn%29%2Cn%E2%88%88N%2A%2C%E8%8B%A5%E9%9B%86%E5%90%88M%3D%7Bn%7Cbn%E2%89%A5%CE%BB%2Cn%E2%88%88N%2A%7D%E6%81%B0%E6%9C%894%E4%B8%AA%E5%85%83%E7%B4%A0%2C%E6%B1%82%E5%AE%9E%E6%95%B0%CE%BB%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4.)
已知数列{an}的前n项和为Sn,且a1=1/2,an+1=(n+1)an/2n.(1)求数列{an}的通项公式;(2)设bn=n(2-Sn),n∈N*,若集合M={n|bn≥λ,n∈N*}恰有4个元素,求实数λ的取值范围.
已知数列{an}的前n项和为Sn,且a1=1/2,an+1=(n+1)an/2n.
(1)求数列{an}的通项公式;
(2)设bn=n(2-Sn),n∈N*,若集合M={n|bn≥λ,n∈N*}恰有4个元素,求实数λ的取值范围.
已知数列{an}的前n项和为Sn,且a1=1/2,an+1=(n+1)an/2n.(1)求数列{an}的通项公式;(2)设bn=n(2-Sn),n∈N*,若集合M={n|bn≥λ,n∈N*}恰有4个元素,求实数λ的取值范围.
1.
a(n+1)=(n+1)an/(2n)
a(n+1)/(n+1)=(1/2)(an/n)
[a(n+1)/(n+1)]/(an/n)=1/2,为定值
a1/1=(1/2)/1=1/2,数列{an/n}是以1/2为首项,1/2为公比的等比数列
an/n=1/2ⁿ
an=n/2ⁿ
数列{an}的通项公式为an=n/2ⁿ
2.
Sn=a1+a2+...+an=1/2+2/2²+3/2³+...+n/2ⁿ
Sn /2=1/2²+2/2³+...+(n-1)/2ⁿ+n/2^(n+1)
Sn -Sn /2=Sn /2=1/2+1/2²+...+1/2ⁿ -n/2^(n+1)
=(1/2)(1-1/2ⁿ)/(1-1/2) -n/2^(n+1)
=1- (n+2)/2^(n+1)
Sn=2- (n+2)/2ⁿ
n=1时,b1=1×(2-S1)=1×(2-a1)=1×(2-1/2)=3/2
n=2时,b2=1×(2-S2)=1×(2-2+4/4)=1
n≥2时,
bn=n(2-Sn)=n[2-2+(n+2)/2ⁿ]=n(n+2)/2ⁿ
b(n+1)/bn=[(n+1)(n+3)/2^(n+1)]/[n(n+2)/2ⁿ]
=(n+1)(n+3)/[2n(n+2)]
=(n²+4n+3)/(2n²+4n)
n为正整数,n²+4n+3>0 2n²+4n>0
2n²+4n-(n²+4n+3)=n²-3 n≥2,n²-3>0 2n²+4n>n²+4n+3
0
(1) an=n/(2(n-1))*(n-1)/(2(n-2))......1*a1
=n/(2^n)
(2) sn=2-(n+2)*(1/2)^n
bn=(n^2+2n)*(1/2)^n 35/32