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
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