| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Recurrence relation: evaluate sum |
| Difficulty | Standard +0.8 This question requires students to discover that the recurrence relation produces a periodic sequence (computing a₂ = 0, a₃ = -3, a₄ = 3 reveals period 3), then use this pattern to evaluate sums over 100 terms. While the arithmetic is straightforward once the pattern is found, recognizing the periodicity requires insight beyond routine sequence manipulation, and part (b) adds a layer of interpretation about overlapping sums. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04g Sigma notation: for sums of series |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(a_1=3,\ a_2=0,\ a_3=1.5,\ a_4=3\) | M1 | Uses \(a_{n+1}=\frac{a_n-3}{a_n-2}\) with \(a_1=3\) to generate \(a_2, a_3, a_4\) |
| \(\sum_{r=1}^{100}a_r = 33(4.5)+3\) | M1 | Finds \(a_4=3\) and deduces sum is \(33("3"+"0"+"1.5")+"3"\) |
| \(= 151.5\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sum_{r=1}^{100}a_r + \sum_{r=1}^{99}a_r = (2)(151.5)-3 = 300\) | B1ft | Follow through on periodic function; deduces \(\sum_{r=1}^{100}a_r+\sum_{r=1}^{99}a_r = (2)("151.5")-3=300\) or \("151.5"+(33)("3"+"0"+"1.5")=151.5+148.5=300\) |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $a_1=3,\ a_2=0,\ a_3=1.5,\ a_4=3$ | M1 | Uses $a_{n+1}=\frac{a_n-3}{a_n-2}$ with $a_1=3$ to generate $a_2, a_3, a_4$ |
| $\sum_{r=1}^{100}a_r = 33(4.5)+3$ | M1 | Finds $a_4=3$ and deduces sum is $33("3"+"0"+"1.5")+"3"$ |
| $= 151.5$ | A1 | Correct answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum_{r=1}^{100}a_r + \sum_{r=1}^{99}a_r = (2)(151.5)-3 = 300$ | B1ft | Follow through on periodic function; deduces $\sum_{r=1}^{100}a_r+\sum_{r=1}^{99}a_r = (2)("151.5")-3=300$ or $"151.5"+(33)("3"+"0"+"1.5")=151.5+148.5=300$ |
---\begin{enumerate}
\item A sequence of numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots$ is defined by
\end{enumerate}
$$\begin{aligned}
a _ { 1 } & = 3 \\
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N }
\end{aligned}$$
(a) Find $\sum _ { r = 1 } ^ { 100 } a _ { r }$\\
(b) Hence find $\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }$
\hfill \mbox{\textit{Edexcel Paper 1 Q3 [4]}}