求1*3*5+2*4*6+3*5*7+...n项之和
问题描述：
求1*3*5+2*4*6+3*5*7+...n项之和
答
an=n(n+2)(n+4)=n³+6n²+8n
Sn=[n(n+1)/2]²+6n(n+1)(2n+1)/6+8n(n+1)/2
=n(n+1)(n²+n+8n+4+16)/4
=n(n+1)(n+4)(n+5)/4
答
an=n(n+2)(n+4)=n^3+6n^2+8n
Sn=a1+a2+...+an
=(1^3+6*1^2+8*1)+(2^3+6*2^2+8*2)+...+(n^3+6n^2+8n)
=(1^3+2^3+...+n^3)+6*(1^2+2^2+...+n^2)+8*(1+2+...+n)
=[n(n+1)/2]^2+6*n(n+1)(2n+1)/6+8*n(n+1)/2
=n(n+1)(n+4)(n+5)/4
注：1^3+2^3+...+n^3=[n(n+1)/2]^2
1^2+2^2+...+n^2=n(n+1)(2n+1)/6
1+2+...+n=n(n+1)/2