已知函数f(x)=log3(ax+b)的图象经过点A(2,1)和B(5,2),记an=3f(n),n∈N*.(1)求数列{an}的通项公式;(2)设bn=an2n,Tn=b1+b2+…+bn,,求Tn

题目简介

已知函数f(x)=log3(ax+b)的图象经过点A(2,1)和B(5,2),记an=3f(n),n∈N*.(1)求数列{an}的通项公式;(2)设bn=an2n,Tn=b1+b2+…+bn,,求Tn

题目详情

已知函数f(x)=log3(ax+b)的图象经过点A(2,1)和B(5,2),记an=3f(n),n∈N*
(1)求数列{an}的通项公式;
(2)设bn=
an
2n
Tn=b1+b2+…+bn
,,求Tn
题型:解答题难度:中档来源:烟台三模

答案

(1)由题意得
log3(2a+b)=1
log3(5a+b)=2
,解得
a=2
b=-1
,∴f(x)=log3(2x-1)an=3log3(2n-1)=2n-1,n∈N*
(2)由(1)得bn=class="stub"2n-1
2n
,∴Tn=class="stub"1
21
+class="stub"3
22
+class="stub"5
23
++class="stub"2n-3
2n-1
+class="stub"2n-1
2n
class="stub"1
2
Tn= class="stub"1
22
+class="stub"3
23
++class="stub"2n-5
2n-1
+class="stub"2n-3
2n
+class="stub"2n-1
2n+1
②;
①-②得class="stub"1
2
Tn=class="stub"1
21
+class="stub"2
22
+class="stub"2
23
++class="stub"2
2n-1
+class="stub"2
2n
-class="stub"2n-1
2n+1
=class="stub"1
21
+(class="stub"1
21
+class="stub"1
22
++class="stub"1
2n-2
+class="stub"1
2n-1
)-class="stub"2n-1
2n+1
=class="stub"3
2
-class="stub"1
2n-1
-class="stub"2n-1
2n+1

Tn=3-class="stub"1
2n-2
-class="stub"2n-1
2n
=3-class="stub"2n+3
2n

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