已知等比数列{an}各项均为正数,且2a1+3a2=8,a32=14a2.a6.(1)求数列{an}的前n项和Sn;(2)若数列{bn}满足bn=1log2(Sn+1).log2(Sn+1+1)(n∈

题目简介

已知等比数列{an}各项均为正数,且2a1+3a2=8,a32=14a2.a6.(1)求数列{an}的前n项和Sn;(2)若数列{bn}满足bn=1log2(Sn+1).log2(Sn+1+1)(n∈

题目详情

已知等比数列{an}各项均为正数,且2a1+3a2=8,a32=
1
4
a2a6

(1)求数列{an}的前n项和Sn
(2)若数列{bn}满足bn=
1
log2(Sn+1).log2(Sn+1+1)
(n∈N*)
,求数列{bn}的前n项和Tn
题型:解答题难度:中档来源:不详

答案

(1)∵正项等比数列{an}中,a32=class="stub"1
4
a2•a6=class="stub"1
4
a42
∴q2=
a42
a32
=4,q>0,
∴q=2;
又2a1+3a2=8,即2a1+3a1q=8,
∴a1=1.
∴Sn=
1×(1-2n)
1-2
=2n-1.
(2)∵bn=class="stub"1
log2(Sn+1).log2(Sn+1+1)

=class="stub"1
log2(2n-1+1).log2(2n+1-1+1)

=class="stub"1
log22nlog22n+1

=class="stub"1
n(n+1)

=class="stub"1
n
-class="stub"1
n+1

∴Tn=b1+b2+…+bn
=(1-class="stub"1
2
)+(class="stub"1
2
-class="stub"1
3
)+…+(class="stub"1
n
-class="stub"1
n+1

=1-class="stub"1
n+1

=class="stub"n
n+1

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