Question

设Sn是等差数列｛an｝的前n项和,求证：若正整数m,n,p成等差数列,则Sm/m,Sn/n,Sp/p也成等差数列.

Answer 1

Sn=[(a1+a1+(n-1)d]*n/2=[2a1+(n-1)d)]*n/2

Sm/m={[2a1+(m-1)d)]*m/2}/m=a1+(m-1)d/2

Sn/n=a1+(n-1)d/2

Sp/p=a1+(p-1)d/2

2Sn/n=2a1+(n-1)d

m,n,p成等差数列,2n=m+p

(Sm/m)+(Sp/p)

=a1+(m-1)d/2+a1+(p-1)d/2

=2a1+(m+p-2)d/2

=2a1+(2n-2)d/2

=2a1+(n-1)d

2Sn/n=(Sm/m)+(Sp/p)

Sm/m,Sn/n,Sp/p也成等差数列