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Question 0516
Sigma Notation
2016 Paper 1 Question 6 Variant
Question
Generate new
A sequence
u
0
,
u
1
,
u
2
,
…
{u_0, u_1, u_2, \ldots}
u
0
,
u
1
,
u
2
,
…
is given by
u
0
=
5
and
u
n
=
u
n
−
1
+
n
3
\begin{gather*} u_0 = 5 \textrm{ and} \\ u_n = u_{n-1} + n^3 \end{gather*}
u
0
=
5
and
u
n
=
u
n
−
1
+
n
3
for
n
≥
1.
{n \geq 1.}
n
≥
1.
(i)
Find
u
1
,
u
2
{u_1, u_2}
u
1
,
u
2
and
u
3
.
{u_3.}
u
3
.
[2]
It is further given that
∑
r
=
1
n
r
3
=
n
2
4
(
n
+
1
)
2
.
\sum_{r=1}^n r^3 = {\textstyle \frac{n^2}{4}} (n+1)^2.
r
=
1
∑
n
r
3
=
4
n
2
(
n
+
1
)
2
.
(ii)
By considering
∑
r
=
1
n
(
u
r
−
u
r
−
1
)
,
{\displaystyle \sum_{r=1}^n (u_r - u_{r-1}),}
r
=
1
∑
n
(
u
r
−
u
r
−
1
)
,
find a formula for
u
n
{u_n}
u
n
in terms of
n
.
{n.}
n
.
[3]
Answer
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