设等差数列{an}的前n项和为Sn,等比数列{bn}的前n项和为Tn,已知bn>0(n∈N*),a1=b1=1,a2+b3=a3,S5=5(T3+b2).(Ⅰ)求数列{an},{bn}的通项公式;(Ⅱ

题目简介

设等差数列{an}的前n项和为Sn,等比数列{bn}的前n项和为Tn,已知bn>0(n∈N*),a1=b1=1,a2+b3=a3,S5=5(T3+b2).(Ⅰ)求数列{an},{bn}的通项公式;(Ⅱ

题目详情

设等差数列{an}的前n项和为Sn,等比数列{bn}的前n项和为Tn,已知bn>0(n∈N*),a1=b1=1,a2+b3=a3,S5=5(T3+b2).
(Ⅰ)求数列{an},{bn}的通项公式;
(Ⅱ)求和:
b1
T1T2
+
b2
T2T3
+…+
bn
TnTn+1
题型:解答题难度:中档来源:不详

答案

(Ⅰ)设{an}的公差为d,数列{bn}的公比为q,则
∵a1=b1=1,a2+b3=a3,S5=5(T3+b2),
∴q2=d,1+2d=1+2q+q2,
∴q2-2q=0,
∵q≠0,∴q=2,∴d=4
∴an=4n-3,bn=2n-1;
(Ⅱ)∵
bn
TnTn+1
=
bn+1
qTnTn+1
=class="stub"1
2
class="stub"1
Tn
-class="stub"1
Tn+1

b1
T1T2
+
b2
T2T3
+…+
bn
TnTn+1
=class="stub"1
2
class="stub"1
T1
-class="stub"1
T2
+class="stub"1
T2
-class="stub"1
T3
+…+class="stub"1
Tn
-class="stub"1
Tn+1

=class="stub"1
2
class="stub"1
T1
-class="stub"1
Tn+1
)=class="stub"1
2
(1-class="stub"2
2n+1-1
).

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