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Question
Mathematics
The variance of first n natural numbers is
Q. The variance of first
n
natural numbers is
149
175
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A
2
n
(
n
+
1
)
B
12
(
n
+
1
)
(
n
+
5
)
C
12
(
n
+
1
)
(
n
−
5
)
D
12
(
n
2
−
1
)
Solution:
Variance of first
n
natural numbers
=
n
Σ
n
2
−
(
n
Σ
n
)
2
=
6
n
n
(
n
+
1
)
(
2
n
+
1
)
−
(
2
n
n
(
n
+
1
)
)
2
=
(
n
+
1
)
[
6
2
n
+
1
−
4
(
n
+
1
)
]
=
12
(
n
+
1
)
[
4
n
+
2
−
3
n
−
3
]
=
12
(
n
+
1
)
×
(
n
−
1
)
=
12
n
2
−
1