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China Northern MO
2016 China Northern MO
5
2016 CNMO Grade 11 P5
2016 CNMO Grade 11 P5
Source: 2016 China Northern MO Grade 11, Problem 5
February 25, 2020
algebra
Problem Statement
a
1
=
2
,
a
n
+
1
=
2
n
+
1
a
n
(
n
+
1
2
)
a
n
+
2
n
(
n
∈
Z
+
)
a_1=2,a_{n+1}=\frac{2^{n+1}a_n}{(n+\frac{1}{2})a_n+2^n}(n\in\mathbb{Z}_+)
a
1
=
2
,
a
n
+
1
=
(
n
+
2
1
)
a
n
+
2
n
2
n
+
1
a
n
(
n
∈
Z
+
)
(a) Find
a
n
a_n
a
n
. (b) Let
b
n
=
n
3
+
2
n
2
+
2
n
+
2
n
(
n
+
1
)
(
n
2
+
1
)
a
n
b_n=\frac{n^3+2n^2+2n+2}{n(n+1)(n^2+1)a_n}
b
n
=
n
(
n
+
1
)
(
n
2
+
1
)
a
n
n
3
+
2
n
2
+
2
n
+
2
. Find
S
n
=
∑
i
=
1
n
b
i
S_n=\sum_{i=1}^nb_i
S
n
=
∑
i
=
1
n
b
i
.
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