Question

观察下列各式1*2*3*4+1=5*52*3*4*5+1=11*113*4*5*6+1=19*19……1.请写出一个具有

Answer 1

1

n(n+1)(n+2)(n+3)+1=(n^2+3n+1)^2

证明：

n(n+1)(n+2)(n+3)+1

=[(n(n+3)][(n+1)(n+2)]+1

=(n^2+3n+1-1)(n^2+3n+1+1)+1

=(n^2+3n+1)^2-1+1

=(n^2+3n+1)^2

2

n=2000

2000*2001*2002*2003+1

=(2000^2+2000*3+1)^2

=4006001^2