Question

已知a(n)=1/(n+1)*(n+1),f(n)=[1-a(1)]*…*[1-a(n)],试求f(1),f(2),f(

Answer 1

f(n)=n+2/2*(n+1)

f（1）=3/4 f(2)=2/3=4/6 f(3)=5/8

n=1 f(1)=1+2/2*(1+1)=3/4

假设当n=k f(n)=n+2/2*(n+1)成立

当n=k+1 f(k+1)=[1-a(1)]*…*[1-a(k)]*[1-a(k+1)]=f(k)*[1-1/(k+2)*(k+2)]

=[k+2/2*(k+1)]*[k+3/k+2]*[k+1/k+2]=k+3/2*(k+2)

所以f(n)=n+2/2*(n+1)