2*4+4*6+6*8+…+2n(2n+2)等于多少?

2*4+4*6+6*8+…+2n(2n+2)等于多少?
2*4+4*6+6*8+…+2n(2n+2)等于多少?

2*4+4*6+6*8+…+2n(2n+2)等于多少?
2n(2n+2)
=4n^2+4n
1^2+2^2+3^2+...+n^2=n(n+1)(2n+1)/6
2*4+4*6+6*8+…+2n(2n+2)
=4*1^2+4*1+4*2^2+4*2+4*3^2+4*3+.+4n^2+4n
=4(1^2+2^2+3^2+...+n^2)+4(1+2+3+...+n)
=4*n(n+1)(2n+1)/6+4*(1+n)*n/2
=2n(n+1)(2n+1)/3+2n(n+1)
=2n(n+1)[(2n+1)/3+1]
=2n(n+1)[(2n+1)+3]/3
=2n(n+1)(2n+4)/3
=4n(n+1)(n+2)/3

2*4+4*6+6*8+…+2n(2n+2)=2（1\2+2\3+3\4+...+n\(n+1)）=2[(1-1\2)+（1-1\3）+...+(1-n\(n+1))]=
2[n-(1\2+1\3+...+1\(n+1))]=2(n+1-(11\2+1\3+...+1\(n+1))=2(n+1-ln(n+2))=2(n+1)-1ln(n+2)