题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221223-91149.png)
(1)求a1,a3;
(2)求证:数列{an}为等差数列,并写出其通项公式;
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221223-43451.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221223-22470.png)
(2) an=n-1
(3) 存在唯一正整数数 对(p,q)=(2,3),使b1,bp,bq成等比数列
解析
试题分析:解:(1)令n=1,则a1=S1=
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221223-22470.png)
(2)由
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221223-91149.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221223-89670.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221224-47752.png)
②-①,得
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221224-33330.png)
于是,
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221224-27688.png)
③+④,得
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221224-32636.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221224-59393.png)
又a1=0,a2=1,a2-a1=1,
所以,数列{an}是以0为首项,1为公差的等差数列.
所以,an=n-1. 9分
法二②-①,得
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221224-33330.png)
于是,
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221225-18115.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221225-37830.png)
(3)假设存在正整数数组(p,q),使b1,bp,bq成等比数列,
则lgb1,lgbp,lgbq成等差数列, 10分
于是,
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221225-44413.png)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221226-31765.png)
当p≥3,且p∈N*时,
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221226-22697.png)
故数列{
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221226-16581.png)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221227-38453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221227-47156.png)
综上,存在唯一正整数数 对(p,q)=(2,3),使b1,bp,bq成等比数列. 16分
点评:解决的关键是根据等差数列和等比数列的性质以及定义来求解运用。属于基础题。
核心考点
试题【已知数列{an}中,a2=1,前n项和为Sn,且.(1)求a1,a3;(2)求证:数列{an}为等差数列,并写出其通项公式;(3)设,试问是否存在正整数p,q(】;主要考察你对等差数列等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221205-31665.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-50897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-76326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-52364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-69712.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221207-26550.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221207-33193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221207-33931.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221207-88908.png)
(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-50897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221208-63323.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221208-24165.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-52364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221208-79039.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-52364.png)
(2)已知圆
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221208-19604.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221209-68360.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221209-28445.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221209-95899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221209-91535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221209-75037.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221208-19604.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221207-33931.png)
(3)若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-50897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-76326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-52364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221210-34111.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221206-76326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221210-82688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221210-57181.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221210-20235.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221128-75119.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221129-16536.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221129-76428.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221129-29418.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221129-27242.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221130-18662.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221130-74661.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221130-77736.png)
(1)求证:数列
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221130-72149.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221131-98855.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221131-87321.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221130-77736.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221131-41949.png)
(3)当
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221131-85449.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221132-97559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221132-35239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221129-16536.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221132-19884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221132-19884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221132-66011.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221133-85957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221133-70298.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221132-19884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009221133-72525.png)
(1)写出a1,a2,a3, 并推测a n的表达式;
(2)用数学归纳法证明所得的结论.
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220939-86092.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220939-99007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220939-64702.png)
A.3 | B.4 | C.5 | D.6 |
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220903-99038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220903-12002.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220903-34631.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191009/20191009220904-26584.png)
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