Question

各项均为正数的数列[an],a1=a,a2=b,且对满足m+n=p+q的正整数m,n,p,q都有am+an/(1+am)

Answer 1

由已知 An-1 + An+1 / (1 + An-1)(1 + An+1) = 2An / (1 + An)^2, 将上式两端都乘以-2,再分别加上1,得到 (1 - An-1)/(1 + An-1) * (1 - An+1)/(1 + An+1) = (1-An)^2/(1+An)^2 = ((1-An)/(1+An))^2,n=2,3,4...所以{1-An/1+An}是等比数列,其中首项是1/3,公比是1/3,所以1-An/1+An = 3^n,所以 An = 3^n-1 / 3^n+1 .