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3
2013 PUMaC Algebra B3
2013 PUMaC Algebra B3
Source:
November 22, 2013
Problem Statement
Let
x
1
=
1
/
20
x_1=1/20
x
1
=
1/20
,
x
2
=
1
/
13
x_2=1/13
x
2
=
1/13
, and
x
n
+
2
=
2
x
n
x
n
+
1
(
x
n
+
x
n
+
1
)
x
n
2
+
x
n
+
1
2
x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}
x
n
+
2
=
x
n
2
+
x
n
+
1
2
2
x
n
x
n
+
1
(
x
n
+
x
n
+
1
)
for all integers
n
≥
1
n\geq 1
n
≥
1
. Evaluate
∑
n
=
1
∞
(
1
/
(
x
n
+
x
n
+
1
)
)
\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))
∑
n
=
1
∞
(
1/
(
x
n
+
x
n
+
1
))
.
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