已知公差不为0的等差数列{an}的前n项和为Sn,S3=a4+6,且a1,a4,a13成等比数列. (Ⅰ)求数列{an}的通项公式; (Ⅱ)求数列{1/Sn}的前n项和公式.

问题描述:

已知公差不为0的等差数列{an}的前n项和为Sn,S3=a4+6,且a1,a4,a13成等比数列.
(Ⅰ)求数列{an}的通项公式;
(Ⅱ)求数列{

1
Sn
}的前n项和公式.

(Ⅰ)设公差为d,且d≠0,
∵S3=a4+6,且a1,a4,a13成等比数列
∴3a1+3d=a1+3d+6,(a1+3d)2=a1(a1+12d)
∴a1=3,d=2
∴an=3+2(n-1)=2n+1;
(Ⅱ)Sn=

n(3+2n+1)
2
=n(n+2),∴
1
Sn
=
1
n(n+2)
=
1
2
(
1
n
-
1
n+2
)

∴数列{
1
Sn
}的前n项和为
1
2
(1-
1
3
+
1
2
-
1
4
+
1
3
-
1
5
+…+
1
n
-
1
n+2
)
=
1
2
(1+
1
2
-
1
n+1
-
1
n+2
)

=
3n2+5n
4(n+1)(n+2)