等差数列{an}的前n项和为Sn,a3=20,S3=36,则1S1-1+1S2-1+1S3-1+…+1S15-1=______.-数学

题目简介

等差数列{an}的前n项和为Sn,a3=20,S3=36,则1S1-1+1S2-1+1S3-1+…+1S15-1=______.-数学

题目详情

等差数列{an}的前n项和为Sn,a3=20,S3=36,则
1
S1-1
+
1
S2-1
+
1
S3-1
+…+
1
S15-1
=______.
题型:填空题难度:偏易来源:河南模拟

答案

∵a3=a1+2d=20,S3=3a1+3d=36
∴d=8,a1=4
Sn=4n+
n(n-1)
2
×8=4n2

class="stub"1
Sn-1
=class="stub"1
4n2- 1
=class="stub"1
(2n+1)(2n-1)
=class="stub"1
2
(class="stub"1
2n-1
-class="stub"1
2n+1
)

class="stub"1
S1-1
+class="stub"1
S2-1
+class="stub"1
S3-1
+…+class="stub"1
S15-1

=class="stub"1
1×3
+class="stub"1
3×5
+…+class="stub"1
29×31

=class="stub"1
2
( 1-  class="stub"1
3
+class="stub"1
3
-class="stub"1
5
+…+class="stub"1
29
-class="stub"1
31
)
=class="stub"15
31

故答案为:class="stub"15
31

更多内容推荐