已知数列an的前n项和为Sn,且1/S₁+1/S₂+...+1/Sn=n/n+1(n属于N*)1:求S₁、S₂、Sn2:设bn(n为项数)=(1/2)ˆan(n为项数),数列bn的前n项和为Tn(n为项数),若对一切n属于N*均有Tn
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![已知数列an的前n项和为Sn,且1/S₁+1/S₂+...+1/Sn=n/n+1(n属于N*)1:求S₁、S₂、Sn2:设bn(n为项数)=(1/2)ˆan(n为项数),数列bn的前n项和为Tn(n为项数),若对一切n属于N*均有Tn](/uploads/image/z/5302235-11-5.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E4%B8%941%2FS%26%238321%3B%2B1%2FS%26%238322%3B%2B...%2B1%2FSn%3Dn%2Fn%2B1%EF%BC%88n%E5%B1%9E%E4%BA%8EN%2A%EF%BC%891%EF%BC%9A%E6%B1%82S%26%238321%3B%E3%80%81S%26%238322%3B%E3%80%81Sn2%EF%BC%9A%E8%AE%BEbn%EF%BC%88n%E4%B8%BA%E9%A1%B9%E6%95%B0%EF%BC%89%3D%EF%BC%881%2F2%EF%BC%89%26%23710%3Ban%EF%BC%88n%E4%B8%BA%E9%A1%B9%E6%95%B0%EF%BC%89%2C%E6%95%B0%E5%88%97bn%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BATn%EF%BC%88n%E4%B8%BA%E9%A1%B9%E6%95%B0%EF%BC%89%2C%E8%8B%A5%E5%AF%B9%E4%B8%80%E5%88%87n%E5%B1%9E%E4%BA%8EN%2A%E5%9D%87%E6%9C%89Tn)
已知数列an的前n项和为Sn,且1/S₁+1/S₂+...+1/Sn=n/n+1(n属于N*)1:求S₁、S₂、Sn2:设bn(n为项数)=(1/2)ˆan(n为项数),数列bn的前n项和为Tn(n为项数),若对一切n属于N*均有Tn
已知数列an的前n项和为Sn,且1/S₁+1/S₂+...+1/Sn=n/n+1(n属于N*)
1:求S₁、S₂、Sn
2:设bn(n为项数)=(1/2)ˆan(n为项数),数列bn的前n项和为Tn(n为项数),若对一切n属于N*均有Tn(n为项数)属于(1/m,m²-6m+16/3),求实数m取值范围!
已知数列an的前n项和为Sn,且1/S₁+1/S₂+...+1/Sn=n/n+1(n属于N*)1:求S₁、S₂、Sn2:设bn(n为项数)=(1/2)ˆan(n为项数),数列bn的前n项和为Tn(n为项数),若对一切n属于N*均有Tn
1.
n=1时,1/S1=1/(1+1)=1/2 S1=2
n=2时,1/S1+1/S2=1/2 +1/S2=2/3
1/S2=2/3-1/2=1/6 S2=6
n=1时,S1=2
n≥2时,
1/S1+1/S2+...+1/Sn=n/(n+1) (1)
1/S1+1/S2+...+1/S(n-1)=(n-1)/n (2)
(1)-(2)
1/Sn=n/(n+1)-(n-1)/n=[n²-(n+1)(n-1)]/[n(n+1)]=1/[n(n+1)]=1/(n²+n)
Sn=n²+n
n=1时,S1=1²+1=2,同样满足.
数列{Sn}的通项公式为Sn=n²+n.
2.
n=1时,a1=S1=1+1=2
n≥2时,Sn=n²+n S(n-1)=(n-1)²+(n-1)
an=Sn-S(n-1)=n²+n-(n-1)²-(n-1)=2n
n=1时,a1=2,同样满足
数列{an}的通项公式为an=2n
bn=(1/2)^(an)=(1/2)^(2n)=(1/4)ⁿ
b1=1/4 b(n+1)/bn=[1/4^(n+1)]/(1/4ⁿ)=1/4,为定值.
数列{bn}是以1/4为首项,1/4为公比的等比数列.
Tn=b1+b2+...+bn
=(1/4)×[1-(1/4)ⁿ]/(1-1/4)
=(1/3)(1-1/4ⁿ)
=1/3 -1/(3×4ⁿ)
随n增大,4ⁿ单调递增,3×4ⁿ单调递增,1/(3×4ⁿ)单调递减,1/3-1/(3×4ⁿ)单调递增,当n=1时,1/3 -1/(3×4ⁿ)有最小值1/3 -1/12=1/4,又1/(3×4ⁿ)>0 1/3-1/(3×4ⁿ)