设Sn等比数列{an}的前n项和,且a2=19,S2=49(1)求数列{an}的通项;(2)设bn=nan,求数列{bn}的前n项和Sn.-高二数学

题目简介

设Sn等比数列{an}的前n项和,且a2=19,S2=49(1)求数列{an}的通项;(2)设bn=nan,求数列{bn}的前n项和Sn.-高二数学

题目详情

设Sn等比数列{an}的前n项和,且a2=
1
9
S2=
4
9

(1)求数列{an}的通项;
(2)设bn=
n
an
,求数列{bn}的前n项和Sn
题型:解答题难度:中档来源:不详

答案

(1)设首项为a1,公比为q,由a2=class="stub"1
9
,S2=class="stub"4
9

得:
a1q=class="stub"1
9
a1+a1q=class="stub"4
9

解得:
a1=class="stub"1
3
q=class="stub"1
3

∴an=class="stub"1
3n
;.
(2)∵bn=class="stub"n
an
=class="stub"n
class="stub"1
3n
=n•3n,
∴Sn=3+2×32+3×33+…+n•3n,①
∴3Sn=32+2×33+3×34+…+(n-1)•3n+n•3n+1,②
②-①得2Sn=n•3n+1-(3+32+33+…+3n)=n•3n+1-
3(1-3n)
1-3
=
(2n-1)×3n+1
2
+class="stub"3
2

∴Sn=
(2n-1)×3n+1
4
+class="stub"3
4

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