Higher June 2023 Paper 2R Q23
23 Here are the first three terms of an arithmetic sequence.
\(8p \qquad 7p - 3 \qquad 4p + 2\)
The sum of the first \(n\) terms of the sequence is \(-1914\)
Work out the value of \(n\)
Show your working clearly.
(5)
| Scheme | Marks |
|---|---|
| \((7p - 3) - (8p) = (4p + 2) - (7p - 3)\) oe or \(-p - 3 = -3p + 5\) oe or (\(p\) =) 4 | M1 |
| \(a = 32\) or \(d = -7\) or 32 25 18 | A1 |
| \(\dfrac{n}{2}\left[2(32) + (n - 1)(-7)\right] = -1914\) | M1 |
| \(7n^2 - 71n - 3828\;(= 0)\) oe | A1 |
| Working required Answer: 29 | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for using \(U_2 - U_1 = U_3 - U_2\) or \(U_1 - U_2 = U_2 - U_3\)
Condone missing brackets around \(7p - 3\)
A1: dep on M1
(32 and \(-7\) may be embedded in the \(S_n\) formula or embedded in \(U_n\) formula)
M1: The values of \(a\) and \(d\) must be correct
Condone missing brackets around \(n - 1\)
A1: (can be implied by \(n = 29\) and/or \(n = -\dfrac{132}{7}\))
A1: dep on M2
| Scheme | Marks |
|---|---|
| \(7p - 3 = 8p + d\) \(4p + 2 = 8p + 2d\) \(4p + 2 = 7p - 3 + d\) or \(-3 = p + d\) \(2 = 4p + 2d\) \(5 = 3p + d\) | M1 |
| \(a = 32\) or \(d = -7\) or 32 25 18 | A1 |
| \(\dfrac{n}{2}\left[2(32) + (n - 1)(-7)\right] = -1914\) | M1 |
| \(7n^2 - 71n - 3828\;(= 0)\) oe | A1 |
| Working required Answer: 29 | A1 |
Notes
M1: for using \(U_n = a + (n - 1)d\) to set up 2 equations for \(U_2\) and \(U_3\)
A1: dep on M1
(32 and \(-7\) may be embedded in the \(S_n\) formula or embedded in \(U_n\) formula)
M1: The values of \(a\) and \(d\) must be correct
Condone missing brackets around \(n - 1\)
A1: (can be implied by \(n = 29\) and/or \(n = -\dfrac{132}{7}\))
A1: dep on M2