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ISI Entrance Examination
2023 ISI Entrance UGB
6
6
Part of
2023 ISI Entrance UGB
Problems
(1)
Reciprocal Sums and Boring Bounds
Source: Indian Statistical Institute Entrance UGB 2023/6
5/14/2023
Let
{
u
n
}
n
≥
1
\{u_n\}_{n \ge 1}
{
u
n
}
n
≥
1
be a sequence of real numbers defined as
u
1
=
1
u_1 = 1
u
1
=
1
and
u
n
+
1
=
u
n
+
1
u
n
for all
n
≥
1
.
u_{n+1} = u_n + \frac{1}{u_n} \text{ for all $n \ge 1$.}
u
n
+
1
=
u
n
+
u
n
1
for all
n
≥
1.
Prove that
u
n
≤
3
n
2
u_n \le \frac{3\sqrt{n}}{2}
u
n
≤
2
3
n
for all
n
n
n
.
inequalities
recurrence relation