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Putnam
1956 Putnam
B6
Putnam 1956 B6
Putnam 1956 B6
Source: Putnam 1956
July 5, 2022
Putnam
series
limit
greatest common divisor
Problem Statement
Given
T
1
=
2
,
T
n
+
1
=
T
n
2
−
T
n
+
1
T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1
T
1
=
2
,
T
n
+
1
=
T
n
2
−
T
n
+
1
for
n
>
0.
n>0.
n
>
0.
Prove:(i) If
m
≠
n
,
m \ne n,
m
=
n
,
T
m
T_m
T
m
and
T
n
T_n
T
n
have no common factor greater than
1.
1.
1.
(ii)
∑
i
=
1
∞
1
T
i
=
1.
\sum_{i=1}^{\infty} \frac{1}{T_i }=1.
∑
i
=
1
∞
T
i
1
=
1.
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