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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
(
f
(
r
+
2
)
−
f
(
r
)
)
,
\sum_{r=1}^n \Big( f(r+2) - f(r) \Big),
r
=
1
∑
n
(
f
(
r
+
2
)
−
f
(
r
)
)
,
given that
f
(
0
)
=
5
,
{f(0)=5,}
f
(
0
)
=
5
,
f
(
1
)
=
6
{f(1)=6}
f
(
1
)
=
6
and
f
(
2
)
=
8.
{f(2)=8.}
f
(
2
)
=
8.
Attempt
The answer is of the form
f
(
n
)
+
f
(
n
−
1
)
−
a
{f(n) + f(n-1) - a}
f
(
n
)
+
f
(
n
−
1
)
−
a
f
(
n
+
1
)
+
f
(
n
)
−
a
{f(n+1) + f(n) - a}
f
(
n
+
1
)
+
f
(
n
)
−
a
f
(
n
+
2
)
+
f
(
n
+
1
)
−
a
{f(n+2) + f(n+1) - a}
f
(
n
+
2
)
+
f
(
n
+
1
)
−
a
a
−
f
(
n
+
1
)
−
f
(
n
+
2
)
{a - f(n+1) - f(n+2)}
a
−
f
(
n
+
1
)
−
f
(
n
+
2
)
a
−
f
(
n
)
−
f
(
n
+
1
)
{a - f(n) - f(n+1)}
a
−
f
(
n
)
−
f
(
n
+
1
)
a
−
f
(
n
−
1
)
−
f
(
n
)
{a - f(n-1) - f(n)}
a
−
f
(
n
−
1
)
−
f
(
n
)
where
a
=
{a=}
a
=
Answer
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