Higher January 2022 Paper 2 Q26
26 An arithmetic series has first term \(a\) and common difference \(d\), where \(d\) is a prime number.
The sum of the first \(n\) terms of the series is \(S_n\) and
\(S_m = 39\)
\(S_{2m} = 320\)
Find the value of \(d\) and the value of \(m\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
\((S_m =)\;\dfrac{m}{2}(2a + (m - 1)d) = 39\) oe or \((S_{2m} =)\;\dfrac{2m}{2}(2a + (2m - 1)d) = 320\) oe | M1 |
\((S_m =)\;\dfrac{m}{2}(2a + (m - 1)d) = 39\) oe and \((S_{2m} =)\;\dfrac{2m}{2}(2a + (2m - 1)d) = 320\) oe | M1 |
| eliminate to get \(dm^2 = 242\) oe | M1 |
| \(242 = 2 \times 11 \times 11\) or \(242 = 2 \times 121\) oe | M1 |
| Working required Answer: \(d = 2\) \(m = 11\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: one correct equation for \(S_m\) or \(S_{2m}\) (condone consistent use of \(n\) instead of \(m\))
M1: both equations correct
A1: Dep on M2
Both correct