问题 解答题
已知数列{an}是等差数列,且a1=2,a1+a2+a3=12.
(1)求数列{an}的通项公式;
(2)令bn=
an
2an
,求数列{bn}前n项和
答案

(1)由已知a1=2,a1+a2+a3=12,得a1+a1+d+a1+2d=12,即a1+d=4,

则a2=4,又a1=2,

∴d=2,an=2+2(n-1)=2n;

(2)由(1)知bn=

2n
4n
,设数列{bn}前n项和为Sn,则Sn=
2
4
+
2×2
42
+…+
2n
4n
①,

Sn
4
=
2
16
+
2×2
43
+
2×3
44
+…+
2(n-1)
4n
+
2n
4n+1
②,

又①-②错位相减得:

3
4
Sn=
1
2
+
1
4
-
1
8
+
2
43
(1+
1
4
+…+
2
4n-3
)-
2n
4n+1

=

5
8
+
1
32
×
1-
1
4n-2
1-
1
4
-
2n
4n+1
=
2
3
-
3n+4
4n
,则Sn=
4
3
×
2
3
-
4
3
×
3n+4
4n
=
8
9
-
6n+8
4n

所以数列{bn}前n项和Sn=

8
9
-
6n+8
4n

单项选择题
单项选择题