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Putnam
1947 Putnam
A5
A5
Part of
1947 Putnam
Problems
(1)
Putnam 1947 A5
Source: Putnam 1947
4/3/2022
Let
a
1
,
b
1
,
c
1
a_1 , b_1 , c_1
a
1
,
b
1
,
c
1
be positive real numbers whose sum is
1
,
1,
1
,
and for
n
=
1
,
2
,
…
n=1, 2, \ldots
n
=
1
,
2
,
…
we define
a
n
+
1
=
a
n
2
+
2
b
n
c
n
,
b
n
+
1
=
b
n
2
+
2
a
n
c
n
,
c
n
+
1
=
c
n
2
+
2
a
n
b
n
.
a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.
a
n
+
1
=
a
n
2
+
2
b
n
c
n
,
b
n
+
1
=
b
n
2
+
2
a
n
c
n
,
c
n
+
1
=
c
n
2
+
2
a
n
b
n
.
Show that
a
n
,
b
n
,
c
n
a_n , b_n ,c_n
a
n
,
b
n
,
c
n
approach limits as
n
→
∞
n\to \infty
n
→
∞
and find those limits.
Putnam
Sequences
limit
Sum