假设m>n
Sn=A1+A2+……+An
Sm=A1+A2+……+An+A(n+1)+A(n+2)+……+Am
Sm-Sn=A(n+1)+A(n+2)+……+Am=0 (共m-n项)
从A(n+1)项到Am项也是等差数列
A(n+1)+A(n+2)+……+Am=(A(n+1)+Am)(m-n)/2=0
A(n+1)+Am=0
A(n+1)+Am=An+A(m+1)=A(n-1)+A(m+2)=……=A1+A(m+n)=0
S(m+n)=(A1+A(m+n))(m+n)/2=0
假设m>n
Sn=A1+A2+……+An
Sm=A1+A2+……+An+A(n+1)+A(n+2)+……+Am
Sm-Sn=A(n+1)+A(n+2)+……+Am=0 (共m-n项)
从A(n+1)项到Am项也是等差数列
A(n+1)+A(n+2)+……+Am=(A(n+1)+Am)(m-n)/2=0
A(n+1)+Am=0
A(n+1)+Am=An+A(m+1)=A(n-1)+A(m+2)=……=A1+A(m+n)=0
S(m+n)=(A1+A(m+n))(m+n)/2=0