Higher June 2022 Paper 1R Q25
25 The sum of the first 10 terms of an arithmetic series is 4 times the sum of the first 5 terms of the same series.
The 8th term of this series is 45
Find the first term of this series.
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
\((S_{10} =)\;\dfrac{10}{2}(2a + 9d)\) or \((S_5 =)\;\dfrac{5}{2}(2a + 4d)\) oe or \(a + 7d = 45\) | M1 |
| \(\dfrac{10}{2}(2a + 9d) = 4 \times \dfrac{5}{2}(2a + 4d)\) oe | M1 |
eg \(d = 2a\) oe or \(a = \dfrac{d}{2}\) oe or \(a + 7d = 45\) oe and eg \(10a - 5d = 0\) oe or eg \(\dfrac{10}{2}(2(45 - 7d) + 9d) = 4 \times \dfrac{5}{2}(2(45 - 7d) + 4d)\) oe or \(5d = 10(45 - 7d)\) oe | M1 |
| eg \(a + 7(2a) = 45\) or \(d = 6\) or eg \(70a - 35d = 0\) \(5a + 35d = 225\) + \((75a = 225)\) or \(10a - 5d = 0\) \(10a + 70d = 450\) − \((-75d = -450)\) | M1 |
| Working required Answer: 3 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for a correct expression for the sum of the first 10 terms (\(S_{10}\)) or the first 5 terms (\(S_5\)) or a correct equation for the 8th term
Take 9 as their 10 – 1 and 4 as their 5 – 1 and 7 as their 8 – 1
M1: for a correct equation relating \(S_{10}\) and \(S_5\)
M1: (dep on M1) for \(d\) in terms of \(a\), or vice-versa (must be correct)
or for \(a + 7d = 45\) oe and correctly reducing the equation relating \(S_{10}\) and \(S_5\) to an equation with one term in \(a\) and one term in \(d\) eg \(10a - 5d = 0\) oe
or substituting a correct expression into their correct equation to obtain an equation in just \(d\)
M1: (dep on M2) for a correct equation in just \(a\) or for \(d = 6\) or for a correct method to eliminate \(a\) or \(d\): coefficients of \(a\) or \(d\) the same and correct operation to eliminate selected variable (condone 1 arithmetical error)
A1: Dep on M3