数列2(S2n-Sn)为什么等于Sn+S3n-S2n 怎么来的，求详解

Question

如题所述

Answer 1

设公差为d
2[S(2n)-Sn]=2[a1+a2+...+a(2n)-a1+a2+...+an]
=2[a(n+1)+a(n+2)+...+a(2n)]
=2[a1+nd+a2+nd+...+an+nd]
=2[(a1+a2+...+an)+n²d]
Sn+S(3n)-S(2n)
=a1+a2+...+an+a1+a2+...+a(3n)-[a1+a2+...+a(2n)]
=a1+a2+...+an+a(2n+1)+a(2n+2)+...+a(3n)
=a1+a2+...+an+a1+2nd+a2+2nd+...+an+2nd
=2(a1+a2+...+an)+2n²d
=2[(a1+a2+...+an)+n²d]
2[S(2n)-Sn]=Sn+S(3n)-S(2n)