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All-Russian Olympiad
1978 All Soviet Union Mathematical Olympiad
268
268
Part of
1978 All Soviet Union Mathematical Olympiad
Problems
(1)
ASU 268 All Russian MO 1978 x_n=(1+\sqrt(2)+\sqrt(3))^n , limits
Source:
7/11/2019
Consider a sequence
x
n
=
(
1
+
2
+
3
)
n
x_n=(1+\sqrt2+\sqrt3)^n
x
n
=
(
1
+
2
+
3
)
n
Each member can be represented as
x
n
=
q
n
+
r
n
2
+
s
n
3
+
t
n
6
x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6
x
n
=
q
n
+
r
n
2
+
s
n
3
+
t
n
6
where
q
n
,
r
n
,
s
n
,
t
n
q_n, r_n, s_n, t_n
q
n
,
r
n
,
s
n
,
t
n
are integers. Find the limits of the fractions
r
n
/
q
n
,
s
n
/
q
n
,
t
n
/
q
n
r_n/q_n, s_n/q_n, t_n/q_n
r
n
/
q
n
,
s
n
/
q
n
,
t
n
/
q
n
.
analysis
algebra
Sequence