1：
猜想
Tn=1/(n+1)
证明
T1= 1-1/2 = 1/2 = 1/(1+1)
T(n+1)
= Tn*[1-1/(n+1+1)]
= [1/(n+1)]*[1-1/(n+2))]
= [1/(n+1)]*[(n+1)/(n+2)]
= 1/(n+2)
= 1/[(n+1)+1]
结果成立
命题得证
1：
猜想
Sn=n/(n+1)
证明
S1= 1/1*2 = 1/2 = 1/(1+1)
Sn+1
= Sn+1/(n+1)(n+1+1)
= n/(n+1)+1/(n+1)(n+2)
= [n(n+2)+1]/(n+1)(n+2)
= (n^2+2n+1)/(n+1)(n+2)
= (n+1)^2/(n+1)(n+2)
= (n+1)/(n+2)
= (n+1)/[(n+1)+1]
结果成立
命题得证
3：
(1)
a2
=3a1/(a1+3)
=3/4
a3
=3a2/(a2+3)
=3/5
a4
=3a3/(a3+3)
=1/2(= 3/6)
（2）
猜想
an=3/(n+2)
证明
a1=3/(1+2)=1
a2,a3,a4也满足条件
a(n+1)
=3an/(an+3)
=[3*3/(n+2)]/[3+3/(n+2)]
上下同乘以（n+2）
=9/[3(n+2)+3]
=3/(n+2+1)
=3/[(n+1)+2]
结果成立
命题得证