两个关于数列的简答题,1.数列{an},a1=4,an=an-1+2/[n(n+1)](n≥2),求an=2.数列{an},a1=4,an+1=[1+2/(n+1)]an,求an=

Question

两个关于数列的简答题,
1.数列{an},a1=4,an=an-1+2/[n(n+1)] (n≥2),求an=
2.数列{an},a1=4,an+1=[1+2/(n+1)]an,求an=

Answer 1

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