Sn=na 1+n(n- 1)d/2。 ①
s2n = na 1+2n(2n- 1)d/2。 ②
So:
sn/S2n =[na 1+n(n- 1)d]/[2na 1+2n(2n- 1)d]。 ③
Expand it:
sn/S2n =[a 1+(N- 1)d]/[2a 1+(4n-2)d]。 Omit n.
= 1-(3n- 1)d/[2a 1+(4n-2)d]。 ④
So as long as the following fraction is constant.
And let this constant be k (k≠0, because if k=0, there must be d=0).
Sn/S2n= 1-k .⑤
Then:
(3n- 1)d = k[2a 1+(4n-2)d]。 ⑥
Sorted mobile items:
a 1 =[(3n- 1)/2k-2n+ 1]d .⑦
A 1 and d are constants, and the proportional relationship between them must have nothing to do with n.
So there is
3n/2k-2n=0 (making the term related to n in Equation 7 zero)
Do you understand:
k=3/4
Substitute ⑤ ⑦:
a 1=d/3
Sn/S2n= 1-k= 1/4