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Question 0501b
Sigma Notation
Method of Differences II
Question
Generate new
Find, in terms of
n
,
{n,}
n
,
∑
r
=
2
n
(
u
r
−
2
−
u
r
)
,
\sum_{r=2}^n \Big( u_{r-2} - u_{r} \Big),
r
=
2
∑
n
(
u
r
−
2
−
u
r
)
,
given that
u
0
=
3
,
{u_{0}=3,}
u
0
=
3
,
u
1
=
7
{u_{1}=7}
u
1
=
7
and
u
2
=
4.
{u_{2}=4.}
u
2
=
4.
Attempt
The answer is of the form
u
n
+
u
n
−
1
−
a
{u_{n} + u_{n-1} - a}
u
n
+
u
n
−
1
−
a
u
n
+
1
+
u
n
−
a
{u_{n+1} + u_{n} - a}
u
n
+
1
+
u
n
−
a
u
n
+
2
+
u
n
+
1
−
a
{u_{n+2} + u_{n+1} - a}
u
n
+
2
+
u
n
+
1
−
a
a
−
u
n
+
1
−
u
n
+
2
{a - u_{n+1} - u_{n+2}}
a
−
u
n
+
1
−
u
n
+
2
a
−
u
n
−
u
n
+
1
{a - u_{n} - u_{n+1}}
a
−
u
n
−
u
n
+
1
a
−
u
n
−
1
−
u
n
{a - u_{n-1} - u_{n}}
a
−
u
n
−
1
−
u
n
where
a
=
{a=}
a
=
Answer
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