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Regional Mathematical Olympiad
2018 India Regional Mathematical Olympiad
6
Sequences...
Sequences...
Source: RMO 2018 P6
October 28, 2018
algebra
Problem Statement
Define a sequence
{
a
n
}
n
≥
1
\{a_n\}_{n\geq 1}
{
a
n
}
n
≥
1
of real numbers by
a
1
=
2
,
a
n
+
1
=
a
n
2
+
1
2
,
for
n
≥
1.
a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.
a
1
=
2
,
a
n
+
1
=
2
a
n
2
+
1
,
for
n
≥
1.
Prove that
∑
j
=
1
N
1
a
j
+
1
<
1
\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1
j
=
1
∑
N
a
j
+
1
1
<
1
for every natural number
N
N
N
.
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