利用等比数列的前n项和公式的推导方法,计算Sn=32+54+78+…+2n+12n=______.-数学

题目简介

利用等比数列的前n项和公式的推导方法,计算Sn=32+54+78+…+2n+12n=______.-数学

题目详情

利用等比数列的前n项和公式的推导方法,计算Sn=
3
2
+
5
4
+
7
8
+
+
2n+1
2n
=______.
题型:填空题难度:偏易来源:不详

答案

Sn=class="stub"3
2
+class="stub"5
4
+class="stub"7
8
+
+class="stub"2n+1
2n
,①
class="stub"1
2
Sn
=class="stub"3
4
+class="stub"5
8
+class="stub"7
16
+…+class="stub"2n-1
2n
+class="stub"2n+1
2n+1
,②
①-②,得class="stub"1
2
Sn
=class="stub"3
2
+class="stub"2
4
+class="stub"2
8
+class="stub"2
16
+…+class="stub"2
2n
-class="stub"2n+1
2n+1

=class="stub"1
2
+2(class="stub"1
2
+class="stub"1
4
+class="stub"1
8
+class="stub"1
16
+…+class="stub"1
2n
)-class="stub"2n+1
2n+1

=class="stub"1
2
+2
×
class="stub"1
2
(1-class="stub"1
2n
)
1-class="stub"1
2
-class="stub"2n+1
2n+1

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

∴Sn=5-class="stub"4
2n
-class="stub"2n+1
2n
=5-class="stub"2n+5
2n

故答案为:5-class="stub"2n+5
2n

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