Question

已知数列{an}各项均不为零,且满足关系式an=3an-1/an-1+3(n大于等于2,n属于正整数) (1)当a1=1

Answer 1

an=3a(n-1)/[a(n-1)+3]

1/an=[a(n-1)+3]/[3a(n-1)]=1/a(n-1) +1/3

1/an -1/a(n-1)=1/3,为定值.

1/a1=1/(1/2)=2

数列{1/an}是以2为首项,1/3为公差的等差数列.

1/an=1/a1 +(1/3)(n-1)=2 +n/3 -1/3=(n+5)/3

an=3/(n+5)

n=1时,a1=3/(1+5)=3/6=1/2,同样满足.

数列{an}的通项公式为an=3/(n+5).