解:(1)证明: 由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537237455.gif)
,得
an+1=2
n—
an,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231555376271127.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537643878.gif)
,
∴数列
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537268600.gif)
是首项为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537721452.gif)
,公比为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537736147.gif)
的等比数列.………………3分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537768652.gif)
, 即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537783648.gif)
,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537580495.gif)
…………………………………………………………………………5分
(2)解:假设在数列{
bn}中,存在连续三项
bk-1,
bk,
bk+1(
k∈N*,
k≥2)成等差数列,则
bk-1+
bk+1=2
bk,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537830975.gif)
,
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537846228.gif)
=4
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537861380.gif)
………………………………………………………………7分
若
k为偶数,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537846228.gif)
>0,4
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537861380.gif)
=-4<0,所以,不存在偶数
k,使得
bk-1,
bk,
bk+1成等差数列。…………………………………………………………8分
若
k为奇数,则
k≥3,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537846228.gif)
≥4,而4
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155537861380.gif)
=4,所以,当且仅当
k=3时,
bk-1,
bk,
bk+1成等差数列。
综上所述,在数列{
bn}中,有且仅有连续三项
b2,
b3,
b4成等差数列。…………10分
(3)要使
b1,
br,
bs成等差数列,只需
b1+
bs=2
br,
即3+
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538111418.gif)
=2[
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538126416.gif)
],即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538142609.gif)
, ①
(ⅰ)若
s=
r+1,在①式中,左端
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538204373.gif)
=0,右端
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538220504.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538267652.gif)
,要使①式成立,当且仅当
s为偶数时成立。又
s>
r>1,且
s,
r为正整数,所以,当
s为不小于4的正偶数,且
s=
r+1时,
b1,
br,
bs成等差数列。……………………………………………………………13分
(ⅱ)若
s≥
r+2时,在①式中,左端
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538204373.gif)
≥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538314377.gif)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538345216.gif)
>0,右端
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823155538220504.gif)
≤0,∴当
s≥
r+2时,
b1,
br,
bs不成等差数列。
综上所述,存在不小于4的正偶数
s,且
s=
r+1,使得
b1,
br,
bs成等差数列。…15分