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Baltic Way
1998 Baltic Way
8
Prove binomial sum for the polynomial P
Prove binomial sum for the polynomial P
Source: Baltic Way 1998
January 11, 2011
algebra
polynomial
algebra proposed
Problem Statement
Let
P
k
(
x
)
=
1
+
x
+
x
2
+
…
+
x
k
−
1
P_k(x)=1+x+x^2+\ldots +x^{k-1}
P
k
(
x
)
=
1
+
x
+
x
2
+
…
+
x
k
−
1
. Show that
∑
k
=
1
n
(
n
k
)
P
k
(
x
)
=
2
n
−
1
P
n
(
x
+
1
2
)
\sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right)
k
=
1
∑
n
(
k
n
)
P
k
(
x
)
=
2
n
−
1
P
n
(
2
x
+
1
)
for every real number
x
x
x
and every positive integer
n
n
n
.
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