Edexcel PURE 2024 October — Question 2

Exam BoardEdexcel
ModulePURE
Year2024
SessionOctober
PaperDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeSimple recurrence evaluation
DifficultyStandard +0.8 This is a Further Maths Pure question requiring pattern recognition in a recurrence relation with alternating signs, algebraic manipulation to find the constant k, and insight that the sequence is periodic with period 4 to evaluate a large sum. While systematic, it requires more sophistication than standard A-level sequences questions and tests understanding of periodicity in recurrence relations.
Spec1.04e Sequences: nth term and recurrence relations
  1. A sequence of numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by
$$\begin{gathered} u _ { 1 } = 7 \\ u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k \end{gathered}$$ where \(k\) is a constant.
  1. Show that \(u _ { 5 } = 7\) Given that \(\sum _ { r = 1 } ^ { 4 } u _ { r } = 30\)
  2. find the value of \(k\).
  3. Hence find the value of \(\sum _ { r = 1 } ^ { 150 } u _ { r }\)
Question 2:
AnswerMarks
2(a)u = 7  u = k − 7
1 2B1
u = k − 7  u = k + k − 7  u = k − ( 2 k − 7 )
2 3 4
u = k + ( − k + 7 ) = 7 *
AnswerMarks
5M1A1*
(3)
AnswerMarks
(b)4
u = 3 0  7 + k − 7 + 2 k − 7 − k + 7 = 3 0  k = ...
r
AnswerMarks
r = 1M1
k = 1 5A1cso
(2)
AnswerMarks
(c)150
u =3730+7+"15"−7
r
AnswerMarks
r=1M1
= 1 1 2 5A1cso
(2)

Total 7

AnswerMarks Guidance
n1 2
u
AnswerMarks Guidance
n7 k − 7
Value (with k = 15)7 8
n1 2
u
AnswerMarks Guidance
n7 k + 7
Value (with k = 15)7 22 
Question 2:
--- 2(a) ---
2(a) | u = 7  u = k − 7
1 2 | B1
u = k − 7  u = k + k − 7  u = k − ( 2 k − 7 )
2 3 4
u = k + ( − k + 7 ) = 7 *
5 | M1A1*
(3)
(b) | 4

u = 3 0  7 + k − 7 + 2 k − 7 − k + 7 = 3 0  k = ...
r
r = 1 | M1
k = 1 5 | A1cso
(2)
(c) | 150

u =3730+7+"15"−7
r
r=1 | M1
= 1 1 2 5 | A1cso
(2)
Total 7
n | 1 | 2 | 3 | 4
u
n | 7 | k − 7 | 2k − 7 | 7 – k
Value (with k = 15) | 7 | 8 | 23 | −8
n | 1 | 2 | 3 | 4
u
n | 7 | k + 7 | −7 | k – 7
Value (with k = 15) | 7 | 22 | −7 | 8
\begin{enumerate}
  \item A sequence of numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by
\end{enumerate}

$$\begin{gathered}
u _ { 1 } = 7 \\
u _ { n + 1 } = ( - 1 ) ^ { n } u _ { n } + k
\end{gathered}$$

where $k$ is a constant.\\
(a) Show that $u _ { 5 } = 7$

Given that $\sum _ { r = 1 } ^ { 4 } u _ { r } = 30$\\
(b) find the value of $k$.\\
(c) Hence find the value of $\sum _ { r = 1 } ^ { 150 } u _ { r }$

\hfill \mbox{\textit{Edexcel PURE 2024 Q2}}
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