(1)a(n+1)/an=n/(n+1)
an/a(n-1)=(n-1)/n
.
a2/a1=1/2
所以a(n+1)/an*an/a(n-1) *.*a2/a1
=a(n+1)/a1=1/(n+1)
所以a(n+1)=1/(n+1)
an=1/n
(2)sn=n(2n-1)an
s(n-1)=(n-1)(2n-3)a(n-1)
两式相减得
an=sn-s(n-1)=n(2n-1)an -(n-1)(2n-3)a(n-1)
an*(2n^2-n-1)=a(n-1)*(2n^2-5n+3)
an/a(n-1)=(2n^2-5n+3)/(2n^2-n-1) =(n-1)(2n-3)/[(2n+1)(n-1)]=(2n-3)/(2n+1)
a(n-1)/a(n-2)=(2n-5)/(2n-1)
a(n-2)/a(n-3)=(2n-7)/(2n-3)
.
a3/a2=3/7
a2/a1=1/3
所以an/a(n-1)*a(n-1)/a(n-2)*a(n-2)/a(n-3)*.*a3/a2*a2/a1
=an/a1=3/[(2n-1)(2n-3)]
an=1/[(2n-1)(2n-3)]
(3)a(n+1)=2/3 an +4
令a(n+1)-k=2/3 (an-k)
则k-2/3 k =4
所以k=12
a(n+1)-12=2/3 (an-12)
{a(n+1)-12)}是以公比为2/3 的等比数列
所以a(n+1)-12=(2/3)^n *(a1-12)
a(n+1)=12+(2/3)^n *(-11)
an=12+(2/3)^(n-1)*(-11)