求和:1+11+2+11+2+3+…+11+2+3+…+n=______.-数学

题目简介

求和:1+11+2+11+2+3+…+11+2+3+…+n=______.-数学

题目详情

求和:1+
1
1+2
+
1
1+2+3
+…+
1
1+2+3+…+n
=______.
题型:填空题难度:中档来源:不详

答案

an=class="stub"1
1+2+3+…+n
=class="stub"2
n(n+1)

∴Sn=a1+a2+a3+…+an
=2(class="stub"1
1×2
+class="stub"1
2×3
+class="stub"1
3×4
+…+class="stub"1
n×(n+1)

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

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

故答案:class="stub"2n
n+1

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