已知各项均为正数的数列{an}满足a0=1/2,an=a(n-1)+(1/n^2)*(a(n-1))^2.求证an小于n快,

Question

已知各项均为正数的数列{an}满足a0=1/2,an=a(n-1)+(1/n^2)*(a(n-1))^2.求证an小于n
快,

Answer 1

显然题目中n∈N
当n=1时,a1=1/2+1/4=3/4＜1,命题成立;
假定n=k时,ak＜k
当n=k+1时,a(k+1)=ak+(ak)^2/(k+1)^2
＜k+[k/(k+1)]^2＜k+[(k+1)/(k+1)]^2=k+1,即n=k+1时命题也成立
于是对于n∈N,an＜n.