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Mathematics
Problem
Given that
∑
k
=
1
n
k
=
n
(
n
+
1
)
2
\sum_{k=1}^{n} k = \frac{n(n + 1)}{2}
∑
k
=
1
n
k
=
2
n
(
n
+
1
)
and
∑
k
=
1
n
k
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
\sum_{k=1}^{n} k^2 = \frac{n(n + 1)(2n + 1)}{6}
∑
k
=
1
n
k
2
=
6
n
(
n
+
1
)
(
2
n
+
1
)
, write
∑
k
=
1
n
(
k
+
1
)
(
k
+
2
)
\sum_{k=1}^{n} (k + 1)(k + 2)
∑
k
=
1
n
(
k
+
1
)
(
k
+
2
)
in simplest form. Check your answer in the case when
n
=
10
n = 10
n
=
10
.
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