已知公差不为0的等差数列{an}的首项a1=3,设数列的前项和为Sn,且1a1,1a2,1a4成等比数列.(Ⅰ)求数列

2025-06-21 15:44:41
推荐回答(1个)
回答1:

(Ⅰ)由a1=3且

1
a1
1
a2
1
a4
成等比数列得(
1
a2
)
2
=
1
a1
×
1
a4

(
1
3+d
)
2
=
1
3
×
1
3+3d

解得d=3.
∴数列{an}的通项公式an=3n,
∴Sn=
3n(n+1)
2

(2)∵
1
Sn
=
2
3
1
n
-
1
n+1
),
∴An=
1
S1
+
1
S2
+…+
1
Sn
=
2
3
[(1-
1
2
)+(
1
2
-
1
3
)+…+(
1
n
-
1
n+1
)]
=
2
3
(1-