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{
}的前n项和公式.1 Sn
答
(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(n+2),∴n(3+2n+1) 2
=1 Sn
=1 n(n+2)
(1 2
-1 n
)1 n+2
∴数列{
}的前n项和为1 Sn
(1-1 2
+1 3
-1 2
+1 4
-1 3
+…+1 5
-1 n
)=1 n+2
(1+1 2
-1 2
-1 n+1
)1 n+2
=
.3n2+5n 4(n+1)(n+2)