Higher November 2024 Paper 1 Q19
19 An arithmetic series has first term \(a\) and common difference \(d\)
The sum of the first 30 terms of the arithmetic series is 4395
The sum of the 10th term and the 20th term is 284
Work out the sum of the first 45 terms of the arithmetic series.
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
\(\dfrac{30}{2}\left[2a + (30 - 1)d\right] = 4395\) or \(30a + 435d = 4395\) or \(2a + 29d = 293\) | M1 |
| \(a + (10 - 1)d + a + (20 - 1)d = 284\) or \(a + 9d + a + 19d = 284\) or \(2a + 28d = 284\) or \(a + 14d = 142\) | M1 |
| eg \(\begin{aligned} &2a + 29d = 293 \\ -\;&2a + 28d = 284 \end{aligned}\) \((d = 9)\) or eg \(\begin{aligned} &28a + 406d = 4102 \\ -\;&29a + 406d = 4118 \end{aligned}\) \(((-)\,a = (-)\,16)\) | M1 |
| \(\dfrac{45}{2}\left[2(16) + (45 - 1)9\right]\) | M1 |
| Working required Answer: 9630 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for using \(S_n = \dfrac{n}{2}\left[2a + (n - 1)d\right]\)
M1: for using \(U_n = a + (n - 1)d\) correctly to form an equation
M1: dep on M2 for a correct method to eliminate \(a\) or \(d\):
coefficients of \(a\) or \(d\) the same and correct operator to eliminate selected variable (condone any one arithmetic error)
or
writing \(a\) or \(d\) in terms of the other variable and correctly substituting.
M1: dep on previous M1 for using \(S_n = \dfrac{n}{2}\left[2a + (n - 1)d\right]\) correctly with \(a = 16\) and \(d = 9\)
A1: dep on M2