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其中n=1,2,

Answer 1

1 求a1 a2

a[0]=1/2,a[n]=a[n-1]+a[n-1]^2/n^2

a[1]=a[0]+a[0]^2/1^2=1/2+1/4=3/4

a[2]=3/4+(3/4)^2/2^2=57/64

2 求证 1/a(n-1)-1/ana[n-1]

n^2(a[n]-a[n-1])=a[n-1]^2

所以1/a[n-1]-1/a[n]

=(a[n]-a[n-1])/a[n]a[n-1]

=a[n-1]/n^2[an]