已知数列{an}是等比数列,其中a3=1,且a4,a5+1,a6成等差数列,数列{an/bn}的前n项和Sn=(n-1)2^(n-2)+1(1)求数列{an}、{bn}的通项公式.(2)设数列{bn}的前n项和为Tn,若T3n-Tn≥t对一切正整数n都成立,求实数t
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![已知数列{an}是等比数列,其中a3=1,且a4,a5+1,a6成等差数列,数列{an/bn}的前n项和Sn=(n-1)2^(n-2)+1(1)求数列{an}、{bn}的通项公式.(2)设数列{bn}的前n项和为Tn,若T3n-Tn≥t对一切正整数n都成立,求实数t](/uploads/image/z/935052-60-2.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%98%AF%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E5%85%B6%E4%B8%ADa3%3D1%2C%E4%B8%94a4%2Ca5%2B1%2Ca6%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E6%95%B0%E5%88%97%7Ban%2Fbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CSn%3D%28n-1%292%5E%28n-2%29%2B1%EF%BC%881%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E3%80%81%7Bbn%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F.%EF%BC%882%EF%BC%89%E8%AE%BE%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BATn%2C%E8%8B%A5T3n-Tn%E2%89%A5t%E5%AF%B9%E4%B8%80%E5%88%87%E6%AD%A3%E6%95%B4%E6%95%B0n%E9%83%BD%E6%88%90%E7%AB%8B%2C%E6%B1%82%E5%AE%9E%E6%95%B0t)
已知数列{an}是等比数列,其中a3=1,且a4,a5+1,a6成等差数列,数列{an/bn}的前n项和Sn=(n-1)2^(n-2)+1(1)求数列{an}、{bn}的通项公式.(2)设数列{bn}的前n项和为Tn,若T3n-Tn≥t对一切正整数n都成立,求实数t
已知数列{an}是等比数列,其中a3=1,且a4,a5+1,a6成等差数列,数列{an/bn}的前n项和Sn=(n-1)2^(n-2)+1
(1)求数列{an}、{bn}的通项公式.
(2)设数列{bn}的前n项和为Tn,若T3n-Tn≥t对一切正整数n都成立,求实数t的取值范围.
已知数列{an}是等比数列,其中a3=1,且a4,a5+1,a6成等差数列,数列{an/bn}的前n项和Sn=(n-1)2^(n-2)+1(1)求数列{an}、{bn}的通项公式.(2)设数列{bn}的前n项和为Tn,若T3n-Tn≥t对一切正整数n都成立,求实数t
(1)
a4、a5+1、a6成等差数列,则2(a5+1)=a4+a6
a4=a3q a5=a3q² a6=a3q³ a3=1代入,整理,得
q³-2q²+q-2=0
q²(q-2)+(q-2)=0
(q²+1)(q-2)=0
q²+1恒为正,要等式成立,只有q=2
a1=a3/q²=1/2²=1/4
an=(1/4)×2^(n-1)=2^(n-3)
数列{an}的通项公式为an=2^(n-3).
S1=(1-1)×2^(1-2) +1=1 a1/b1=1 b1=a1=1/4
an/bn=Sn-Sn-1=(n-1)×2^(n-2)+1-(n-2)×2^(n-3)-1=n×2^(n-3)
bn=an/[n×2^(n-3)]=2^(n-3)/[n×2^(n-3)]=1/n
n=1时,b1=1/4,不满足.
数列{bn}的通项公式为
bn=1/4 n=1
1/n n≥2
[T3(n+3)-T(n+1)]-(T3n-Tn)
=[1/4+1/2+1/3+...+1/(3n)+1/(3n+1)+1/(3n+2)+1/(3n+3)]-[1/4+1/2+1/3+...+1/n+1/(n+1)]
-[1/4+1/2+1/3+...+1/(3n)]+(1+1/2+1/3+...+1/n)
=1/(3n+1)+1/(3n+2)+1/(3n+3)-1/(n+1)
>1/(3n+3)+1/(3n+3)+1/(3n+3)-1/(n+1)
=1/(n+1)-1/(n+1)=0
T3(n+3)-T(n+1)>T3n-Tn
即随n增大,T3n-Tn单调递增,当n=1时,T3n-Tn取得最小值.
T3-T1=(1/4+1/2+1/3)-(1/4)=1/2+1/3=5/6
要不等式T3n-Tn≥t对于一切正整数n恒成立,只要t≤5/6.