Question

数列An中,a1=a,an=(an-1)/na(n-1)+1.an=?

Answer 1

a=0时,a(n)=0.

a不为0时,a(n+1)=a(n)/[(n+1)a(n)+1],

1/a(n+1) = [(n+1)a(n) + 1]/a(n) = 1/a(n) + (n+1) = 1/a(n) + [(n+1)(n+2)-n(n+1)]/2,

1/a(n+1) - (n+1)(n+2)/2 = 1/a(n) - n(n+1)/2

{1/a(n) - n(n+1)/2}是首项为1/a(1) - 1 = 1/a - 1,的常数数列.

1/a(n) - n(n+1)/2 = 1/a - 1,

1/a(n) = 1/a + n(n+1)/2 - 1 = [2+n(n+1)a-2a]/(2a),

a(n) = 2a/[2-2a+n(n+1)a]