设等差数列为An,首项为A1,公差=d,所有项=n,则
Ar=A1+(r-1)d
An=A1+(n-1)d
末r项的首项=A(n-r+1)=A1+(n-r)d
S前r=(A1+Ar)*r/2=a
2A1+(r-1)d=2a/r.(1)
S末r=[A(n-r+1)+An]*r/2=b
2A1+(2n-r-1)d=2b/r.(2)
(A1+An)*n/2=S
2A1+(n-1)d=2S/n.(3)
(2)-(1),d(n-r)=(b-a)/r.(4)
(3)-(1),d(n-r)=2S/n-2a/r.(5)
(5)-(4),2S/n-2a/r-(b-a)/r=0
n=2Sr/(a+b)
答：项数n=2Sr/(a+b)