9.已知数列?an?满足:a1?1,an?1?an?an2?n?1?2?n?N?.
?(Ⅰ)证明:
an?11?1?; 2an?n?1?2?n?1??an?1?n?1.
n?3(Ⅱ)证明:
an*n?N10.已知数列?an?满足:a1?1,an?1?an?.(), 2(n?1)证明:当n?N时, (Ⅰ)
*2an?11; ?1?an(n?1)2(Ⅱ)
2(n?1)?an?1?n?1 n?3.
11.已知数列{an}满足a1?2an2?,an?1?,n?N.
3?an5(1)求a2,并求数列{1}的通项公式; an6221. (1?()n)?Sn?5313(2)设{an}的前n项的和为Sn,求证:
12.数列?an?满足a1?1,an?1(1)证明:an?1?an; (2)证明:
n2an?2(n?N?) n?1aa1a21?????n?n?2?; a2a3an?1n(3)证明:an?
1. 4
3annxx13.对任意正整数,设是关于的方程?nx?1的最大实数根
(1)求证:n?an?an?1?n?2 (2)当n?4时,对任意的正整数m,
n?m?n?an?m?an?2(n?m?n)
2(3)设数列{1n2n} 2的前n项和为Sn,求证:ln(1?)?Sn?1?an33