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=
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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=
,设数列{bn}前n项和为Sn,则Sn=2n 4n
+2 4
+…+2×2 42
①,2n 4n
=Sn 4
+2 16
+2×2 43
+…+2×3 44
+2(n-1) 4n
②,2n 4n+1
又①-②错位相减得:
Sn=3 4
+1 2
-1 4
+1 8
(1+2 43
+…+1 4
)-2 4n-3 2n 4n+1
=
+5 8
×1 32
-1- 1 4n-2 1- 1 4
=2n 4n+1
-2 3
,则Sn=3n+4 6×4n
×4 3
-2 3
×4 3
=3n+4 6×4n
-8 9 6n+8 9×4n
所以数列{bn}前n项和Sn=
-8 9 6n+8 9×4n