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Question 0501b
Sigma Notation
Method of Differences II
Question
Generate new
Find, in terms of
n
,
{n,}
n
,
∑
r
=
1
n
(
(
r
+
2
)
3
−
r
3
)
.
\sum_{r=1}^n \Big( (r+2)^3 - r^3 \Big).
r
=
1
∑
n
(
(
r
+
2
)
3
−
r
3
)
.
Attempt
The answer is of the form
n
3
+
(
n
−
1
)
3
−
a
{n^3 + (n-1)^3 - a}
n
3
+
(
n
−
1
)
3
−
a
(
n
+
1
)
3
+
n
3
−
a
{(n+1)^3 + n^3 - a}
(
n
+
1
)
3
+
n
3
−
a
(
n
+
2
)
3
+
(
n
+
1
)
3
−
a
{(n+2)^3 + (n+1)^3 - a}
(
n
+
2
)
3
+
(
n
+
1
)
3
−
a
a
−
(
n
+
1
)
3
−
(
n
+
2
)
3
{a - (n+1)^3 - (n+2)^3}
a
−
(
n
+
1
)
3
−
(
n
+
2
)
3
a
−
n
3
−
(
n
+
1
)
3
{a - n^3 - (n+1)^3}
a
−
n
3
−
(
n
+
1
)
3
a
−
(
n
−
1
)
3
−
n
3
{a - (n-1)^3 - n^3}
a
−
(
n
−
1
)
3
−
n
3
where
a
=
{a=}
a
=
Answer
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