| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Arithmetic progression with parameters |
| Difficulty | Standard +0.8 This question requires students to find k using the common difference property, determine the first term and common difference, derive the sum formula S_n = n²(k+2), and recognize it as a perfect square. It combines algebraic manipulation with proof, going beyond routine application of formulas to require insight about the structure of the result. |
| Spec | 1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((5k-10)-(2k) = (7k-14)-(5k-10) \Rightarrow k = \ldots\) | M1 | 2.1 — Complete method to find \(k\) |
| \(3k - 10 = 2k - 4 \Rightarrow k = 6\) | A1 | 1.1b — Correct method gives \(k=6\) |
| \(k=6 \Rightarrow T_2=12,\ T_3=20,\ T_4=28\), so \(d=8,\ a=4\) | M1 | 2.2a — Uses \(k\) to deduce \(d\) and first term (\(\neq T_2\)) |
| \(S_n = \frac{n}{2}(2(4)+(n-1)(8))\) | M1 | 1.1b — Applies \(S_n = \frac{n}{2}(2a+(n-1)d)\) with their \(a \neq T_2\) and \(d\) |
| \(= \frac{n}{2}(8+8n-8) = 4n^2 = (2n)^2\), which is a square number | A1 | 2.1 — Correctly shows sum is \((2n)^2\) with appropriate conclusion |
## Question 11:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(5k-10)-(2k) = (7k-14)-(5k-10) \Rightarrow k = \ldots$ | M1 | 2.1 — Complete method to find $k$ |
| $3k - 10 = 2k - 4 \Rightarrow k = 6$ | A1 | 1.1b — Correct method gives $k=6$ |
| $k=6 \Rightarrow T_2=12,\ T_3=20,\ T_4=28$, so $d=8,\ a=4$ | M1 | 2.2a — Uses $k$ to deduce $d$ and first term ($\neq T_2$) |
| $S_n = \frac{n}{2}(2(4)+(n-1)(8))$ | M1 | 1.1b — Applies $S_n = \frac{n}{2}(2a+(n-1)d)$ with their $a \neq T_2$ and $d$ |
| $= \frac{n}{2}(8+8n-8) = 4n^2 = (2n)^2$, which is a square number | A1 | 2.1 — Correctly shows sum is $(2n)^2$ with appropriate conclusion |
---\begin{enumerate}
\item The second, third and fourth terms of an arithmetic sequence are $2 k , 5 k - 10$ and $7 k - 14$ respectively, where $k$ is a constant.
\end{enumerate}
Show that the sum of the first $n$ terms of the sequence is a square number.
\hfill \mbox{\textit{Edexcel Paper 2 Q11 [5]}}