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Vietnam Team Selection Test
2011 Vietnam Team Selection Test
4
4
Part of
2011 Vietnam Team Selection Test
Problems
(1)
a_n is a sequence with a_0=1, a_1=3, a_{n+2}=...
Source: Vietnamese TST 2011 P4
4/27/2011
Let
⟨
a
n
⟩
n
≥
0
\langle a_n\rangle_{n\ge 0}
⟨
a
n
⟩
n
≥
0
be a sequence of integers satisfying
a
0
=
1
,
a
1
=
3
a_0=1, a_1=3
a
0
=
1
,
a
1
=
3
and
a
n
+
2
=
1
+
⌊
a
n
+
1
2
a
n
⌋
∀
n
≥
0.
a_{n+2}=1+\left\lfloor \frac{a_{n+1}^2}{a_n}\right\rfloor \ \ \forall n\ge0.
a
n
+
2
=
1
+
⌊
a
n
a
n
+
1
2
⌋
∀
n
≥
0.
Prove that
a
n
⋅
a
n
+
2
−
a
n
+
1
2
=
2
n
a_n\cdot a_{n+2}-a_{n+1}^2=2^n
a
n
⋅
a
n
+
2
−
a
n
+
1
2
=
2
n
for every natural number
n
.
n.
n
.
floor function
induction
number theory proposed
number theory