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Chennai Mathematical Institute B.Sc. Entrance Exam
2023 CMI B.Sc. Entrance Exam
2
g(m + n) = g(m) + mn(m + n) + g(n)
g(m + n) = g(m) + mn(m + n) + g(n)
Source:
May 9, 2023
CMI 2023
functional equation
Problem Statement
Solve for
g
:
Z
+
→
Z
+
g : \mathbb{Z}^+ \to \mathbb{Z}^+
g
:
Z
+
→
Z
+
such that
g
(
m
+
n
)
=
g
(
m
)
+
m
n
(
m
+
n
)
+
g
(
n
)
g(m + n) = g(m) + mn(m + n) + g(n)
g
(
m
+
n
)
=
g
(
m
)
+
mn
(
m
+
n
)
+
g
(
n
)
Show that
g
(
n
)
g(n)
g
(
n
)
is of the form
∑
i
=
0
d
c
i
n
i
\sum_{i=0}^{d} {c_i n^i}
∑
i
=
0
d
c
i
n
i
\\ and find necessary and sufficient conditions on
d
d
d
and
c
0
,
c
1
,
⋯
,
c
d
c_0, c_1, \cdots , c_d
c
0
,
c
1
,
⋯
,
c
d
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