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Mathematics
Problem
For a positive integer
n
n
n
, let
T
(
n
)
=
∑
k
=
1
n
(
3
k
3
+
k
2
)
T(n)=\displaystyle\sum_{k=1}^{n}\bigl(3k^{3}+k^{2}\bigr)
T
(
n
)
=
k
=
1
∑
n
(
3
k
3
+
k
2
)
. Which of the following closed-form expressions equals
T
(
n
)
T(n)
T
(
n
)
? a)
3
[
n
(
n
+
1
)
2
]
2
−
n
(
n
+
1
)
(
2
n
+
1
)
6
3\left[\dfrac{n(n+1)}{2}\right]^{2}-\dfrac{n(n+1)(2n+1)}{6}
3
[
2
n
(
n
+
1
)
]
2
−
6
n
(
n
+
1
)
(
2
n
+
1
)
b)
3
[
n
(
n
+
1
)
2
]
2
+
n
(
n
+
1
)
(
2
n
−
1
)
6
3\left[\dfrac{n(n+1)}{2}\right]^{2}+\dfrac{n(n+1)(2n-1)}{6}
3
[
2
n
(
n
+
1
)
]
2
+
6
n
(
n
+
1
)
(
2
n
−
1
)
c)
3
[
n
(
n
+
1
)
2
]
2
+
n
(
n
+
1
)
(
2
n
+
1
)
6
3\left[\dfrac{n(n+1)}{2}\right]^{2}+\dfrac{n(n+1)(2n+1)}{6}
3
[
2
n
(
n
+
1
)
]
2
+
6
n
(
n
+
1
)
(
2
n
+
1
)
d)
[
n
(
n
+
1
)
2
]
2
+
n
(
n
+
1
)
(
2
n
+
1
)
6
\left[\dfrac{n(n+1)}{2}\right]^{2}+\dfrac{n(n+1)(2n+1)}{6}
[
2
n
(
n
+
1
)
]
2
+
6
n
(
n
+
1
)
(
2
n
+
1
)
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