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Byju's Answer
Standard XII
Mathematics
Vn Method
Sum the serie...
Question
Sum the series
n
.1
+
(
n
−
1
)
.2
+
(
n
−
2
)
.3
+
.
.
.
.1
.
n
.
is
n
(
n
+
1
)
(
n
+
2
)
6
.If you think this is true write 1 otherwise write 0
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Solution
T
r
=
(
n
−
r
+
1
)
r
=
(
n
+
1
)
r
−
r
2
∴
S
n
=
(
n
+
1
)
n
∑
r
=
1
r
−
n
∑
r
=
1
r
2
=
(
n
+
1
)
n
(
n
+
1
)
2
−
n
(
n
+
1
)
(
2
n
−
1
)
6
=
n
(
n
+
1
)
(
n
+
2
)
6
Thus the answer is 1.
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1
Similar questions
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If
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Q.
If
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(1)
n
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(
n
−
1
)
n
+
1
+
n
(
n
−
1
)
2
!
(
n
−
2
)
n
+
1
−
⋯
=
1
2
n
(
n
+
1
)
!
;
(2)
n
n
−
(
n
+
1
)
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n
−
1
)
n
+
(
n
+
1
)
n
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n
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2
)
n
−
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the series in each case being extended to
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terms; and
(3)
1
n
−
n
2
n
+
n
(
n
−
1
)
1
⋅
2
3
n
−
⋯
=
(
−
1
)
n
n
!
;
(4)
(
n
+
p
)
n
−
n
(
n
+
p
−
1
)
n
+
n
(
n
−
1
)
2
!
(
n
+
p
−
2
)
n
−
⋯
=
n
!
;
the series in the last two cases being extended to
n
+
1
terms.
Q.
The sum of the series
1
×
n
+
2
(
n
−
1
)
+
3
(
n
−
2
)
+
…
…
+
(
n
−
1
)
×
2
+
n
×
1
is
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