The value of Co +3C1, +5C2 + ..... +(2n +1)Cn, is equal to
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Answer:
Consider the following
(1+x
2
)
n
=1+
n
C
1
x
2
+
n
C
2
x
4
+
n
C
3
x
6
...
n
C
n
x
2n
Multiplying both sides with x, we get
x(1+x
2
)
n
=x+
n
C
1
x
3
+
n
C
2
x
5
+
n
C
3
x
7
...
n
C
n
x
2n+1
Differentiating both sides with respect to x, we get
[(1+x
2
)
n
+nx(2x)(1+x
2
)
n−1
]
x=1
=1+3
n
C
1
+5
n
C
2
+...(2n+1)
n
C
n
Hence
1+3
n
C
1
+5
n
C
2
+...(2n+1)
n
C
n
=[(1+x
2
)
n
+nx(2x)(1+x
2
)
n−1
]
x=1
=2
n
+2n(2
n−1
)
=2
n
+n2
n
=2
n
(n+1)
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