2010安徽高考数学理科第20题

Question

请高手们来解答

Answer 1

1/(a_1 a_2 )+1/(a_2 a_3 )+1/(a_3 a_4 )+……+1/(a_n a_(n+1) )=n/(a_1 a_(n+1) )
求证{a_n}是等差数列
解：1/(a_1 a_2 )+1/(a_2 a_3 )+1/(a_3 a_4 )+……+1/(a_n a_(n+1) )=n/(a_1 a_(n+1) ) （1）
故有
1/(a_1 a_2 )+1/(a_2 a_3 )+1/(a_3 a_4 )+……+1/(a_(n-1) a_n )=(n-1)/(a_1 a_n ) （2）
（1）-（2）得
a_1=na_n-(n-1) a_(n+1) (3)
类推得
a_1=（n+1)a_(n+1)-n a_(n+2) (4)
联立（3）-（4）
得2a_(n+1)=a_n+a_(n+2)
证毕
而充分条件只要设出等差数列一般形式，带入等式两边相等即可

Answer 2

23/8