| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Periodic Sequences |
| Difficulty | Standard +0.8 This is a Further Maths question on periodic sequences requiring understanding of the periodicity condition (a₃ = a₁), solving a quadratic equation, and summing a periodic sequence. While it involves multiple steps, the techniques are relatively standard for FM students: substitution, algebraic manipulation, and recognizing the pattern in summation. The conceptual demand is moderate—understanding what 'order 2' means and applying it systematically. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
| Answer | Marks | Guidance |
|---|---|---|
| \(a_2 = 5 - p(a_1) = 5 - pa_1\); \(a_3 = 5 - p(a_2) = 5 - p(5 - pa_1) = 5 - 5p + 4p^2\) | M1 | Applies the sequence formula \(a_{n+1} = 5 - pa_n\) to find \(a_2\) and \(a_3\) |
| A1 | Both are correct: \(a_2 = 5 - 4p\) and \(a_3 = 5 - 5p + 4p^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| As the sequence is of order 2, \(a_1 = a_3\), therefore \(4 = 5 - 5p + 4p^2 \Rightarrow p = 1, p = \frac{1}{4}\), so we choose \(p = 1\) | M1 | Sets \(a_3 = 4\) and attempts to find the value of \(p\) |
| A1 | \(p = 1\) |
| Answer | Marks |
|---|---|
| $\sum_{r=1}^{21} a_r = a_1 + a_2 + a_3 + a_4 + \cdots + a_{19} + a_{20} + a_{21} = 4 + 1 + 4 + 1 + \cdots + 4 + 1 + 4 |
**Part a:** Write down an expression for $a_2$ and $a_3$
| $a_2 = 5 - p(a_1) = 5 - pa_1$; $a_3 = 5 - p(a_2) = 5 - p(5 - pa_1) = 5 - 5p + 4p^2$ | M1 | Applies the sequence formula $a_{n+1} = 5 - pa_n$ to find $a_2$ and $a_3$ |
| --- | --- | --- |
| | A1 | Both are correct: $a_2 = 5 - 4p$ and $a_3 = 5 - 5p + 4p^2$ |
**Part b:** Find the value of $p$
| As the sequence is of order 2, $a_1 = a_3$, therefore $4 = 5 - 5p + 4p^2 \Rightarrow p = 1, p = \frac{1}{4}$, so we choose $p = 1$ | M1 | Sets $a_3 = 4$ and attempts to find the value of $p$ |
| --- | --- | --- |
| | A1 | $p = 1$ |
**Part c:** Find $\sum_{r=1}^{21} a_r$
| $\sum_{r=1}^{21} a_r = a_1 + a_2 + a_3 + a_4 + \cdots + a_{19} + a_{20} + a_{21} = 4 + 1 + 4 + 1 + \cdots + 4 + 1 + 4\begin{enumerate}
\item A sequence $a _ { 1 } , a _ { 2 } a _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$a _ { n + 1 } = 5 - p a _ { n } \quad n \geq 1$$
where $p$ is a constant.\\
Given that
\begin{itemize}
\item $a _ { 1 } = 4$
\item the sequence is a periodic sequence of order 2.\\
a. Write down an expression for $a _ { 2 }$ and $a _ { 3 }$.\\
b. Find the value of $p$.\\
c. Find $\sum _ { r = 1 } ^ { 21 } a _ { r }$
\end{itemize}
\hfill \mbox{\textit{Edexcel PMT Mocks Q13 [6]}}