设an(n=2,3,4,…)是(3-x)n的展开式中含x的系数,则1a2+3a3+9a4+…+32007a2009的值等于()A.2008B.2009C.40162009D.40182008-数学

题目简介

设an(n=2,3,4,…)是(3-x)n的展开式中含x的系数,则1a2+3a3+9a4+…+32007a2009的值等于()A.2008B.2009C.40162009D.40182008-数学

题目详情

设an(n=2,3,4,…)是(3-
x
)n
的展开式中含x的系数,则
1
a2
+
3
a3
+
9
a4
+…+
32007
a2009
的值等于(  )
A.2008B.2009C.
4016
2009
D.
4018
2008
题型:单选题难度:偏易来源:不详

答案

由题意可得 an=
C2n
•3n-2,∴class="stub"1
a2
+class="stub"3
a3
+class="stub"9
a4
+…+
32007
a2009
=class="stub"1
C22
+class="stub"1
C23
+class="stub"1
C24
+…+class="stub"1
C22009
 
=2(class="stub"1
1×2
+class="stub"1
2×3
+class="stub"1
3×4
+…+class="stub"1
2008×2009
)=2(1-class="stub"1
2
+class="stub"1
2
-class="stub"1
3
+class="stub"1
3
-class="stub"1
4
+…+class="stub"1
2008
-class="stub"1
2009

=2(1-class="stub"1
2009
)=class="stub"4016
2009

故选C.

更多内容推荐