June 2024 Paper 1 Q10
10
(a) An arithmetic sequence has 300 terms.
The first term of the sequence is \(-7\) and the last term is 32
Find the sum of the 300 terms. [2 marks]
(b) A school holds a raffle at its summer fair.
There are nine prizes.
The total value of the prizes is £1260
The values of the prizes form an arithmetic sequence.
The top prize has the highest value, and the bottom prize has the least value.
The value of the top prize is six times the value of the bottom prize.
Find the value of the top prize. [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(n = 300, a = -7\) and \(l = 32\) Into \(S_n = \dfrac{n}{2}(a + l)\) Or Substitutes \(n = 300, a = -7\) and \(d = \dfrac{39}{299} = \dfrac{3}{23}\) into \(S_n = \dfrac{n}{2}\left(2a + (n-1)d\right)\) Condone \(n\) = 299 or 301 and \(d =\) AWRT 0.13 | M1 | 3.1a |
| Obtains 3750 | A1 | 1.1b |
| (2) |
Typical solution
\[S_{300} = \frac{300}{2}(-7 + 32)\]\[= 3750\]| Scheme | Marks | AO |
|---|---|---|
| Forms an equation using \(S_9 = 1260\) Might see \(\dfrac{9}{2}(2a + 8d) = 1260 \Rightarrow a + 4d = 140\) | M1 | 3.4 |
| Forms an equation using the relationship between the highest and least values. eg \(a + 8d = 6a\) or \(l = 6a\) OE Might see \(l = \dfrac{1}{6}a\) which may indicate the candidate is correctly working from the highest term to the lowest term. | M1 | 3.4 |
| Obtains and solves an equation in one variable having formed one equation using \(S_9 = 1260\) OR used the relationship between the highest and least values. | M1 | 3.1a |
| Obtains £240 Must have correct units. CAO | A1 | 3.2a |
| (4) | ||
| (6 marks) |
Typical solution
\[\frac{9}{2}(a + l) = 1260 \Rightarrow a + l = 280\]\[l = 6a\]\[7a = 280\]\[a = 40,\ l = 240\]Value of top prize = £240