已知数列{an}的前n项和为Sn,若{Sn/an}是公差为d的等差数列,则{an}为等差数列时,d值为题难过程多的话俺酌情加分,俺实在做不出啊.

Question

已知数列{an}的前n项和为Sn,若{Sn/an}是公差为d的等差数列,则{an}为等差数列时,d值为 题难过程多的话俺酌情加分,俺实在做不出啊.

Answer 1

由题设 S1/a1=a1/a1=1
S2/a2=a1/a2+1=1+d,所以 a1=da2
S3/a3=(a1+a2+a3)/a3=(d+1)a2/a3+1=1+2d,因S1/a1有意义,a1≠0,所以d≠0
因此 a3=(d+1)a2/(2d)
若{an}为等差序列,则 a3-a2=a2-a1,即2a2=a1+a3
所以2a2=da2+(d+1)/(2d)a2,因S2/a2有意义,所以a2≠0,
可得4d=2d^2+d+1 即 2d^2-3d+1=2d(d-1)-(d-1)=(2d-1)(d-1)=0,所以d=0.5或d=1
当d=0.5时,序列为{a1,2a1,...,na1} ;当d=1时,序列为{a1,a1,...,a1},经检验均符合题设.