Question

已知f(x)=3x/(x+3),数列{an}满足an=f(an-1) (n>1,a1≠0）求证①{1/an}是等差数列

Answer 1

证明1,an=f(an-1)=3a(n-1)/(a(n-1)+3)所以1/an={a(n-1)+3}/3a(n-1)

即1/an-1/a(n-1)=1/3 所以{1/an}s是等差数列,公差为1/3

(2）由1得1/an=1/a1+(n-1)/3=4+n/3-1/3=11/3+n/3=(11+n)/3

所以,an=3/(11+n) 即a40=3/(11+40)=1/17