| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Real-world AP: find n satisfying a condition |
| Difficulty | Easy -1.2 This is a straightforward application of arithmetic sequences requiring only recognition of the pattern (constant difference of £0.25), writing the nth term formula, and solving a quadratic inequality from the sum formula. All steps are routine with no novel insight needed, making it easier than average A-level content. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| VIAV SIHI NI III IM ION OC | VIIN SIHI NI III M M O N OO | VIAV SIHI NI IIIIM I ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| There is a common difference (no common ratio) and so an arithmetic series should be used. | B1 | Identifies common difference between consecutive terms, or ratio is not the same; must state arithmetic series/sequence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \((u_n) = 5 + (n-1)"d"\) or \((u_n) = 5 + n"d"\) | M1 | Attempts general term for A.S. with \(n\) or \(n-1\) used |
| \((u_n) = 5 + 0.25(n-1)\) | A1 | Correct expression; accept equivalents e.g. \(4.75 + 0.25n\) |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(S_n = \frac{n}{2}(2 \times 5 + (n-1) \times 0.25) \geqslant 350\) | M1A1 | Uses sum formula with their \(a\) and \(d\); any inequality or equality symbol |
| \(\Rightarrow 0.25n^2 + 9.75n \geqslant 700 \Rightarrow (n+19.5)^2 - 19.5^2 \geqslant 2800\) | M1 | Forms and solves 3-term quadratic; any valid method including calculator |
| 37 weeks | A1 | |
| (4) | For attempts via listing, send to review |
## Question 1:
### Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| There is a common difference (no common ratio) and so an arithmetic series should be used. | **B1** | Identifies common difference between consecutive terms, or ratio is not the same; must state arithmetic series/sequence |
### Part (a)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $(u_n) = 5 + (n-1)"d"$ or $(u_n) = 5 + n"d"$ | **M1** | Attempts general term for A.S. with $n$ or $n-1$ used |
| $(u_n) = 5 + 0.25(n-1)$ | **A1** | Correct expression; accept equivalents e.g. $4.75 + 0.25n$ |
| | **(3)** | |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $S_n = \frac{n}{2}(2 \times 5 + (n-1) \times 0.25) \geqslant 350$ | **M1A1** | Uses sum formula with their $a$ and $d$; any inequality or equality symbol |
| $\Rightarrow 0.25n^2 + 9.75n \geqslant 700 \Rightarrow (n+19.5)^2 - 19.5^2 \geqslant 2800$ | **M1** | Forms and solves 3-term quadratic; any valid method including calculator |
| 37 weeks | **A1** | |
| | **(4)** | For attempts via listing, send to review |
---\begin{enumerate}
\item Adina is saving money to buy a new computer. She saves $\pounds 5$ in week $1 , \pounds 5.25$ in week 2 , $\pounds 5.50$ in week 3 and so on until she has enough money, in total, to buy the computer.
\end{enumerate}
She decides to model her savings using either an arithmetic series or a geometric series.\\
Using the information given,\\
(a) (i) state with a reason whether an arithmetic series or a geometric series should be used,\\
(ii) write down an expression, in terms of $n$, for the amount, in pounds ( $\pounds$ ), saved in week $n$.
Given that the computer Adina wants to buy costs $\pounds 350$\\
(b) find the number of weeks it will take for Adina to save enough money to buy the computer.\\
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VIAV SIHI NI III IM ION OC & VIIN SIHI NI III M M O N OO & VIAV SIHI NI IIIIM I ION OC \\
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\hfill \mbox{\textit{Edexcel P2 2021 Q1 [7]}}