设正项数列{an}的前n项和是Sn,若{an}和{Sn}都是等差数列,且公差相等,则a1+d=______.-数学

题目简介

设正项数列{an}的前n项和是Sn,若{an}和{Sn}都是等差数列,且公差相等,则a1+d=______.-数学

题目详情

设正项数列{an}的前n项和是Sn,若{an}和{
Sn
}都是等差数列,且公差相等,则a1+d=______.
题型:填空题难度:中档来源:奉贤区二模

答案

由题意知数列{an}的首项为a1,公差为d.
因为数列{an}的前n项和是Sn,
所以
S1
=
a1
S2
=
2a1+d
S3
=
3a1+3d

又{
Sn
}也是公差为d的等差数列,
S2
=
2a1+d
=
a1
+d
,两边平方得:2a1+d=a1+2d
a1
+d2

S3
=
3a1+3d
=
a1
+2d
,两边平方得:3a1+3d=a1+4d
a1
+4d2

②-①得:a1=-2d+2d
a1
+3d2
③,
把③代入①得:d(2d-1)=0.
所以d=0或d=class="stub"1
2

当d=0时,a1=0,不合题意,
当d=class="stub"1
2
时,代入③解得a1=class="stub"1
4

所以a1+d=class="stub"1
4
+class="stub"1
2
=class="stub"3
4

故答案为class="stub"3
4

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