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Putnam
1950 Putnam
B5
B5
Part of
1950 Putnam
Problems
(1)
Putnam 1950 B5
Source:
5/25/2022
Answer either (i) or (ii).(i) Given that the sequence whose
n
n
n
th term is
(
s
n
+
2
s
n
+
1
)
(s_n + 2s_{n + 1})
(
s
n
+
2
s
n
+
1
)
converges, show that the sequence
{
s
n
}
\{ s_n \}
{
s
n
}
converges also.(ii) A plane varies so that it includes a cone of constant value equal to
π
a
3
/
3
\pi a^3 / 3
π
a
3
/3
with the surface the equation of which in rectangular coordinates is
2
x
y
=
z
2
.
2xy = z^2.
2
x
y
=
z
2
.
Find the equation of the envelope of the various positions of this plane.State the result so that it applies to a general cone (that is, conic surface) of the second order.
Putnam