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Mathematics
Problem
If
(
1
+
x
)
n
=
C
0
+
C
1
x
+
C
2
x
2
+
⋯
+
C
n
x
n
(1+x)^n = C_0 + C_1 x + C_2 x^2 + \dots + C_n x^n
(
1
+
x
)
n
=
C
0
+
C
1
x
+
C
2
x
2
+
⋯
+
C
n
x
n
, prove that
3
C
0
+
3
2
C
1
2
+
3
3
C
2
3
+
⋯
+
3
n
C
n
n
+
1
=
3
n
+
1
C
0
4
n
+
1
−
1
3C_0 + 3^2 \frac{C_1}{2} + 3^3 \frac{C_2}{3} + \dots + \frac{3^n C_n}{n+1} = \frac{3^{n+1} C_0}{4^{n+1}-1}
3
C
0
+
3
2
2
C
1
+
3
3
3
C
2
+
⋯
+
n
+
1
3
n
C
n
=
4
n
+
1
−
1
3
n
+
1
C
0
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