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1977 IMO Longlists
25
Prove the binomial identity [ILL 1974]
Prove the binomial identity [ILL 1974]
Source:
January 11, 2011
induction
Problem Statement
Prove the identity
(
z
+
a
)
n
=
z
n
+
a
∑
k
=
1
n
(
n
k
)
(
a
−
k
b
)
k
−
1
(
z
+
k
b
)
n
−
k
(z+a)^n=z^n+a\sum_{k=1}^n\binom{n}{k}(a-kb)^{k-1}(z+kb)^{n-k}
(
z
+
a
)
n
=
z
n
+
a
k
=
1
∑
n
(
k
n
)
(
a
−
kb
)
k
−
1
(
z
+
kb
)
n
−
k
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