In fact, it is not difficult, but there is a key point: take the reciprocal of the equal sign of an+ 1=3an/(3+an) on both sides and simplify it. If you try, you will easily get a arithmetic progression formula, and then the following one will be simple.

By mathematical induction:

When n=2, a2 is equal to 3/7.

When n=3, a3 is equal to 3/8.

When n=4, a4 equals 3/9.

......

So guess an= 3/(n+5)

Proved by mathematical induction as follows:

Verify the correctness of the beginning: when n=2, it is obviously correct.

Suppose that n=x is correct and ax=3/(x+5).

Then n=x+ 1 ax+ 1=3ax/(3+ax) given by known conditions.

Substituting for, for example, ax=3/(x+5) and then simplifying, you can get

ax+ 1=3/(x+ 1+5)

The conjecture is proved to be true, and the general term holds for any positive number.