数列S12+322+523…+2n-12n的前n项和为______.-数学

题目简介

数列S12+322+523…+2n-12n的前n项和为______.-数学

题目详情

数列S
1
2
+
3
22
+
5
23
…+
2n-1
2n
的前n项和为 ______.
题型:填空题难度:中档来源:不详

答案

Sn=class="stub"1
2
+class="stub"3
22
+class="stub"5
23
…+class="stub"2n-1
2n

class="stub"1
2
Sn=class="stub"1
22
+class="stub"3
23
+class="stub"5
24
+…+class="stub"2n-2
2n
+class="stub"2n-1
2n+1

两式相减得class="stub"1
2
Sn=class="stub"1
2
+class="stub"2
22
+class="stub"2
23
+…+class="stub"2
2n
-class="stub"2n-1
2n+1
=class="stub"1
2
+
class="stub"1
4
(1-class="stub"1
2n-1
)
1-class="stub"1
2
-class="stub"2n-1
2n+1
=class="stub"3
2
-class="stub"2n-1
2n+1

∴Sn=3-class="stub"2n+3
2n

故答案为:3-class="stub"2n+3
2n

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