在等差数列｛an｝中,S(m)=S(n),则S(m+n)的值为?

Question

在等差数列｛an｝中,S(m)=S(n),则S(m+n)的值为?

Answer 1

m=n时 S(m+n)=S(2n)=S(2m)
m不等于n
S(m)=ma1+m(m-1)d/2
S(n)=na1+n(n-1)d/2
S(m)=S(n)
ma1+m(m-1)d/2=na1+n(n-1)d/2
(m-n)a1+(m^2-m-n^2+n)d/2=0
(m-n)a1+(m-n)(m+n-1)d/2=0
a1+(m+n-1)d/2=0
(m+n-1)d/2=-a1
S(m+n)=(m+n)a1+(m+n)(m+n-1)d/2
=(m+n)(a1-a1)
=0