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CHMMC problems
2014 CHMMC (Fall)
7
2014 Fall Team #7
2014 Fall Team #7
Source:
March 26, 2022
algebra
Problem Statement
Let
P
(
x
)
=
∑
k
=
1
n
(
x
3
k
+
x
−
3
k
−
1
)
,
Q
(
x
)
=
∑
k
=
1
n
(
x
3
k
+
x
−
3
k
+
1
)
.
P(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}- 1), Q(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}+ 1).
P
(
x
)
=
k
=
1
∑
n
(
x
3
k
+
x
−
3
k
−
1
)
,
Q
(
x
)
=
k
=
1
∑
n
(
x
3
k
+
x
−
3
k
+
1
)
.
Given that
P
(
x
)
Q
(
x
)
=
∑
k
=
−
2
⋅
3
n
2
⋅
3
n
a
k
x
k
,
P(x)Q(x) =\sum^{2\cdot 3^n}_{k=-2\cdot 3^n} a_kx^k,
P
(
x
)
Q
(
x
)
=
k
=
−
2
⋅
3
n
∑
2
⋅
3
n
a
k
x
k
,
Compute
∑
k
=
0
3
n
a
k
\sum^{3^n}_{k=0}a_k
∑
k
=
0
3
n
a
k
in terms of
n
n
n
.
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