题目
题型:不详难度:来源:
(Ⅰ)若数列{bn}的前n项和为Sn,且a1=b1=d=2,S3<5b2+a88-180,求整数q的值
(Ⅱ)在(Ⅰ)的条件下,试问数列{bn}中是否存在一项bk,使得b,k恰好可以表示为该数列中连续P(P∈N,P≥2)项和?请说明理由。
(Ⅲ)若b1=ar,b2=as≠ar, b3=at(其中t>s>r,且(s—r)是(t—r)的约数)求证:数列{bn}中每一项都是数列{an}中的项.
答案
解析
(1)若数列{bn}的前n项和为Sn,且a1=b1=d=2,S3<5b2+a88-180,借助于通项公式得到q的值。
(Ⅱ)在(Ⅰ)的条件下,假设数列{bn}中存在一项bk,使得b,k恰好可以表示为该数列中连续P(P∈N,P≥2)项和,然后推理证明。
(Ⅲ)若b1=ar,b2=as≠ar, b3=at(其中t>s>r,且(s—r)是(t—r)的约数),要证明数列{bn}中每一项都是数列{an}中的项,只要分析通项公式的特点可以得到。
(Ⅰ)由题意知an=2n,bn=2·
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-71924.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
b1+b2+b3<a88+5b2-180
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
解得1<q<3,q为值数.
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
(Ⅱ)假设数列{bn}中存在一项bk满足bk=bm+bm+1+……bm+p—1
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
又bk=2k=bm+bm+1=2m+2m+1+2m+p—1=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-28095.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
(Ⅲ)由b1=ar得b2=b1q=arq=as=ar+(s—r)d,则d=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120249-27504.png)
又b3=b1q2=ar.q2=at=ar+(t—r)d
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120249-14854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120249-73187.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120249-34746.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-67913.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
故q=
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120249-73187.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
对于数列{bn}中任一项bi(这里只讨论i>3的情形),
有bi=arqi—1= ar+ar(qi—1—1)= ar+ ar(q—1)(1+q+…+qi—2)
= ar+d(s—r)(1+q+…+qi—2)=ar+[((s—r)(1+q+…+qi+2)+1)—1]d
由于(s—r)(1+q+…+qi—2)+1为正整数
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120248-77337.png)
核心考点
试题【(本小题满分14分)已知数列{an}是以d为公差的等差数列,数列{bn}是以q为公比的等比数列(Ⅰ)若数列{bn}的前n项和为Sn,且a1=b1=d=2,S3<】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
在各项均为正数的等比数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120222-49813.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120223-98197.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120223-93563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120223-36653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120223-91731.png)
(Ⅰ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120222-49813.png)
(Ⅱ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120223-40191.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120224-53563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120224-31733.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120224-97276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120208-49574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120209-22820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120209-92493.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120209-32887.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120209-85703.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120210-42462.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120028-96875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120028-94146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120028-44477.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120028-87678.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120029-27875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120028-44477.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120029-17459.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120029-36651.png)
(Ⅰ)求数列
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120028-96875.png)
(Ⅱ)已知
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120029-14761.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120029-27875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120030-71184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120030-35286.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120030-58573.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010120030-56819.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010115951-74548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010115951-69966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010115952-13013.png)
A.![]() | B.6 | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010115757-39695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191010/20191010115758-46770.png)
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