(1)
a1=2
a2=a1²-1×a1+1=4-2+1=3
a3=a2²-2×a2+1=9-6+1=4
a4=a3²-3×a3+1=16-12+1=5
a5=a4²-4×a4+1=25-20+1=6
(2)
猜想：an=n+1
證：
假設(shè)當(dāng)n=k (k∈N+且k≥1)時(shí),ak=k+1
則當(dāng)n=k+1時(shí),
a(k+1)=ak²-k×ak+1
=(k+1)²-k(k+1)+1
=k²+2k+1-k²-k+1
=k+2
=(k+1)+1
同樣成立.
綜上,得an=n+1
用數(shù)學(xué)歸納法很容易證明的.