题目

(理)已知数列{an}中,a1=1,nan+1=2(a1+a2+…+an)(n∈N*).(1)求a2,a3,a4;(2)求数列{an}的通项an;(3)设数列{bn}满足b1=,bn+1=bn2+bn,求证:bn＜1(n≤k).(文)已知O为坐标原点,点E、F的坐标分别为(-1,0)和(1,0),动点P满足=4.(1)求动点P的轨迹C的方程;(2)过E点作直线与C相交于M、N两点,且,求直线MN的方程. 答案：(理)解：(1)a2=2,a3=3,a4=4. (2)nan+1=2(a1+a2+…+an),①(n-1)an=2(a1+a2+…+an-1),②①-②得nan+1-(n-1)an=2an,即nan+1=(n+1)an,, 所以an=a1·=n(n≥2).所以an=n(n∈N*).(3)由(2)得b1=,bn+1=bn2+bn＞bn＞bn-1＞…＞b1＞0,所以{bn}是单调递增数列,故要证bn＜1(n≤k)只需证bk＜1.若k=1,则b1=＜1显然成立, 若k≥2,则bn+1=bn2+bn＜bnbn+1+bn,所以-＞-.因此,. 所以bk＜Which of the following festival is NOT a traditional Chinese festival? ( )A.Spring festivalB.Mid-autumn DayC.Tree Planting DayD.Duanwu Festival