设数列{an}满足a1=2,an+1=an+1/an(n=1,2,3.),证明:an>根号下(2n+1).急用

问题描述:

设数列{an}满足a1=2,an+1=an+1/an(n=1,2,3.),证明:an>根号下(2n+1).急用

证明:(开方即 1/2 次方,用^表示)n=1时,A_1 = 4^(1/2) = 2 > (2*1 + 1)^(1/2) = 3^(1/2);假设n=k时,A_k > (2k + 1)^(1/2);则n=(k+1)时,A_(k+1) = A_k + 1/A_k;欲证结果,只需证(A_k + 1/A_k)^2 > (2k + 3);而(A_k ...