| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Periodic or repeating sequence |
| Difficulty | Moderate -0.8 This is a straightforward question requiring students to compute a few terms to identify the period (order 2: 3, 5, 3, 5...), then use this pattern to sum 85 terms. The recurrence relation is simple, the periodicity is immediately obvious after calculating two terms, and the summation requires only basic arithmetic (42 complete cycles plus one extra term). This is easier than average A-level content as it involves minimal conceptual depth and is largely computational. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04f Sequence types: increasing, decreasing, periodic1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a_1 = 3,\ a_2 = 5,\ a_3 = 3\ ...\) | B1 | At least \(a_2=5\) and \(a_3=3\) must be stated; algebraic approach e.g. \(a_{n+2}=8-(8-a_n)=a_n\) allowed |
| Period \(= 2\) | B1 | Allow "second order", "repeats every 2 numbers"; \(\pm 2\) is B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\displaystyle\sum_{n=1}^{85} a_n = 42\times(3+5)+3\) or equivalent correct method | M1 | e.g. \(\dfrac{84}{2}\times3+42\times5+3\), or \(43\times3+42\times5\), or \(\dfrac{85}{2}\times8-1\); AP formula scores M0 |
| \(= 339\) | A1 | Correct answer only scores both marks |
## Question 3:
### Part (a)(i) and (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_1 = 3,\ a_2 = 5,\ a_3 = 3\ ...$ | B1 | At least $a_2=5$ and $a_3=3$ must be stated; algebraic approach e.g. $a_{n+2}=8-(8-a_n)=a_n$ allowed |
| Period $= 2$ | B1 | Allow "second order", "repeats every 2 numbers"; $\pm 2$ is B0 |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\displaystyle\sum_{n=1}^{85} a_n = 42\times(3+5)+3$ or equivalent correct method | M1 | e.g. $\dfrac{84}{2}\times3+42\times5+3$, or $43\times3+42\times5$, or $\dfrac{85}{2}\times8-1$; AP formula scores M0 |
| $= 339$ | A1 | Correct answer only scores both marks |
---\begin{enumerate}
\item A sequence of terms $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$\begin{aligned}
a _ { 1 } & = 3 \\
a _ { n + 1 } & = 8 - a _ { n }
\end{aligned}$$
(a) (i) Show that this sequence is periodic.\\
(ii) State the order of this periodic sequence.\\
(b) Find the value of
$$\sum _ { n = 1 } ^ { 85 } a _ { n }$$
\hfill \mbox{\textit{Edexcel Paper 2 2022 Q3 [4]}}