数列{an}的前n项和Sn=n2+n,设数列{bn},bn=2an.(1)求数列{bn}的前n项和Tn;(2)求Rn=a1b1+a2b2+…+anbn.-数学

题目简介

数列{an}的前n项和Sn=n2+n,设数列{bn},bn=2an.(1)求数列{bn}的前n项和Tn;(2)求Rn=a1b1+a2b2+…+anbn.-数学

题目详情

数列{an}的前n项和Sn=n2+n,设数列{bn},bn=2an
(1)求数列{bn}的前n项和Tn
(2)求Rn=a1b1+a2b2+…+anbn
题型:解答题难度:中档来源:不详

答案

(1)∵n=1时,a1=S1=2,
n≥2时,an=Sn-Sn-1=n2+n-(n-1)2-(n-1)=2n,
∴an=2n,∴bn=22n=4n,
Tn=b1+b2+…+bn=41+42+…+4n
=
4(1-4n)
1-4
=class="stub"4
3
(4n-1)

(2)Rn=a1b1+a2b2+…+anbn=2×41+4×42+…+2n×4n…①
两边同乘以4得:4Rn=2×42+4×43+…+2n×4n+1…②
①-②得:-3Rn=2×41+2×42+2×43+…+2×4n-2n×4n+1
=2×
4(1-4n)
1-4
-2n×4n+1=(class="stub"8
3
-8n)4n-class="stub"8
3

∴Rn=(class="stub"8
3
n-class="stub"8
9
)4n+class="stub"8
9

更多内容推荐