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Contests
National and Regional Contests
Kosovo Contests
Kosovo Team Selection Test
2015 Kosovo Team Selection Test
1
1
Part of
2015 Kosovo Team Selection Test
Problems
(1)
Prove that for every n
Source: Kosovo TST 2015 Q1
3/30/2015
a)Prove that for every n,natural number exist natural numbers a and b such that
(
1
−
2
)
n
=
a
−
b
2
(1-\sqrt{2})^n=a-b\sqrt{2}
(
1
−
2
)
n
=
a
−
b
2
and
a
2
−
2
b
2
=
(
−
1
)
n
a^2-2b^2=(-1)^n
a
2
−
2
b
2
=
(
−
1
)
n
b)Using first equation prove that for every n exist m such that
(
2
−
1
)
n
=
m
−
m
−
1
(\sqrt{2}-1)^n=\sqrt{m}-\sqrt{m-1}
(
2
−
1
)
n
=
m
−
m
−
1
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