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Byju's Answer
Standard XI
Mathematics
Sum of binomial coefficients of odd numbered terms
If C0,C1,C2...
Question
If
C
0
,
C
1
,
C
2
,
C
3
,
.
.
.
,
C
n
denote the binomial coefficients in the expansion of
(
1
+
x
)
n
, then prove that
1
2
C
1
+
2
2
C
2
+
3
2
C
3
+
.
.
.
.
+
n
2
C
n
=
n
(
n
+
1
)
2
n
−
2
Open in App
Solution
1
2
C
1
+
2
2
C
2
+
3
2
C
3
+
.
.
.
.
+
n
2
C
n
∑
n
r
=
1
r
2
n
C
r
=
∑
n
r
=
1
r
2
n
r
n
−
1
C
r
−
1
using the formula
n
C
r
=
n
r
n
−
1
C
r
−
1
=
n
∑
n
r
=
1
r
n
−
1
C
r
−
1
=
n
∑
n
r
=
1
{
(
r
−
1
)
+
1
}
n
−
1
C
r
−
1
=
n
∑
n
r
=
1
(
r
−
1
)
n
−
1
C
r
−
1
+
n
∑
n
r
=
1
n
−
1
C
r
−
1
=
n
∑
n
r
=
1
(
n
−
1
)
n
−
2
C
r
−
2
+
n
∑
n
r
=
1
n
−
1
C
r
−
1
=
n
(
n
−
1
)
(
0
+
n
−
2
C
0
+
n
−
2
C
1
+
n
−
2
C
2
+
.
.
.
+
n
−
2
C
n
−
2
)
+
n
(
n
−
1
C
0
+
n
−
1
C
1
+
n
−
1
C
2
+
.
.
.
+
n
−
1
C
n
−
1
)
=
n
(
n
−
1
)
2
n
−
2
+
n
.
2
n
−
1
=
n
(
n
+
1
)
2
n
−
2
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0
Similar questions
Q.
If
C
0
,
C
1
,
C
2
,
C
3
,
.
.
.
,
C
n
denote the binomial coefficients in the expansion of
(
1
+
x
)
n
, then
1
2
.
C
1
+
2
2
.
C
2
+
3
2
.
C
3
+
.
.
.
+
n
2
.
C
n
=
Q.
If
C
0
,
C
1
,
C
2
,
…
,
C
n
denote the binomial coefficients in the expansion of
(
1
+
x
)
n
,
then value of
1
2
⋅
C
1
+
2
2
⋅
C
2
+
3
2
⋅
C
3
+
…
+
n
2
C
n
is
Q.
If
C
0
,
C
1
,
C
2
,
.
.
.
C
n
are the binomial coefficients in the expansion of
(
1
+
x
)
n
then prove that:
C
1
C
0
+
2
C
2
C
1
+
3
C
3
C
2
+
.
.
.
.
+
n
C
n
C
n
−
1
=
n
(
n
+
1
)
2
Q.
Prove that:
1
2
.
C
1
+
2
2
.
C
2
+
3
2
.
C
3
+
.
.
.
.
.
n
2
.
C
n
=
n
(
n
+
1
)
2
n
−
2
Q.
If
c
0
,
c
1
,
c
2
,
.
.
.
.
c
n
denote the coefficients in the expansion of
(
1
+
x
)
n
, prove that
c
1
c
0
+
2
c
2
c
1
+
3
c
3
c
2
+
.
.
.
.
.
+
n
c
n
c
n
−
1
=
n
(
n
+
1
)
2
.
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