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Mathematics
Mathematics
Problem
Suppose the function
g
n
(
x
)
=
x
2
n
+
1
+
a
n
x
+
b
n
g_n(x) = x^{2n+1} + a_n x + b_n
g
n
(
x
)
=
x
2
n
+
1
+
a
n
x
+
b
n
(
n
∈
N
n \in \mathbb{N}
n
∈
N
) satisfies the equation
∫
−
1
1
(
p
x
+
q
)
g
n
(
x
)
d
x
=
0
\int_{-1}^{1} (px+q)g_n(x)dx = 0
∫
−
1
1
(
p
x
+
q
)
g
n
(
x
)
d
x
=
0
for all linear functions
(
p
x
+
q
)
(px+q)
(
p
x
+
q
)
, then a.
a
n
=
b
n
=
0
a_n = b_n = 0
a
n
=
b
n
=
0
b.
b
n
=
0
;
a
n
=
−
3
2
n
+
3
b_n = 0; a_n = -\frac{3}{2n+3}
b
n
=
0
;
a
n
=
−
2
n
+
3
3
c.
a
n
=
0
;
b
n
=
−
3
2
n
+
3
a_n = 0; b_n = -\frac{3}{2n+3}
a
n
=
0
;
b
n
=
−
2
n
+
3
3
d.
a
n
=
3
2
n
+
3
;
b
n
=
−
3
2
n
+
3
a_n = \frac{3}{2n+3}; b_n = -\frac{3}{2n+3}
a
n
=
2
n
+
3
3
;
b
n
=
−
2
n
+
3
3
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