∴a2=a 12a 1+ 1= 13,a3=a22a2+ 1= 15,a4=a32a3+ 1= 17.？ ... three o'clock

(2) We can guess an = 1 2n from (1). 1 ...4 points

Prove by mathematical induction:

I) when n= 1, a 1 = 12× 1? 1 = 1, so the conjecture holds when n= 1. ... five points

Ii) Suppose that the conjecture holds when n=k(k∈N*), that is, AK = 12k? 1,

When n=k+ 1, AK+1= ak2ak+1=12k? 12? 12k？ 1+ 1= 12k？ 12+2k？ 12k？ 1 = 12k+ 1 = 12(k+ 1)？ 1

So the conjecture also holds when n=k+ 1.

It can be seen from I) and II) that this conjecture holds for any n∈N*.

So an = 12n? 1 ...8 points