Higher November 2021 Paper 1 Q21
21 The first two terms of a quadratic sequence are 10 and 17
Here is some information about the sequence.
Work out an expression for the \(n\)th term of the sequence. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: using the left hand values | ||
| (\(a\) =) \(6 \div 2\) or (\(a\) =) 3 | M1 | implied by \(3n^2 \ldots\) |
| \(3 \times\) their \(3 + b = 7\) or \(b = -2\) | M1dep | oe \(3n^2 - 2n \ldots\) implies M1M1 |
| \(3 +\) their \(-2 + c = 10\) or \(c = 9\) | M1dep | oe |
| \(3n^2 - 2n + 9\) | A1 | SC1 30 and 49 as the next two terms |
| Alternative method 2: subtracting \(3n^2\) to get a linear sequence | ||
| (\(a\) =) \(6 \div 2\) or (\(a\) =) 3 | M1 | implied by \(3n^2 \ldots\) |
| \(10 -\) their \(3 \times 1^2\) or 7 and \(17 -\) their \(3 \times 2^2\) or 5 or \(b = -2\) | M1dep | oe using any two terms \(3n^2 - 2n \ldots\) implies M1M1 |
| (their 5 − their 7) (\(\times\) 1) \(+ c = 7\) or \(-2\) (\(\times\) 1) \(+ c = 7\) or \(c = 9\) | M1dep | oe equation using any term |
| \(3n^2 - 2n + 9\) | A1 | SC1 30 and 49 as the next two terms |
| Alternative method 3: simultaneous equations | ||
| Simultaneous equations leading to a fully correct method to work out \(a\) or \(b\) or \(a = 3\) or \(b = -2\) | M1 | eg \(a + b + c = 10\) and \(4a + 2b + c = 17\) and \(9a + 3b + c = 30\) and \(3a + b = 7\) and \(5a + b = 13\) and \(2a = 6\) and (\(a\) =) 3 implied by \(3n^2 \ldots\) or \(\ldots -2n \ldots\) |
| Substitutes for \(a\) or \(b\) in one or two of the simultaneous equations with fully correct method to work out the other value | M1dep | eg \(3 \times\) their \(3 + b = 7\) or \(b = -2\) \(3n^2 - 2n \ldots\) implies M1M1 |
| Substitutes for \(a\) & \(b\) to work out \(c\) or \(c = 9\) | M1dep | any term eg \(3 - 2 + c = 10\) |
| \(3n^2 - 2n + 9\) | A1 | SC1 30 and 49 as the next two terms |
| Alternative method 4: Using the ‘0th’ term to get \(c\) | ||
| (\(a\) =) \(6 \div 2\) or (\(a\) =) 3 | M1 | implied by \(3n^2 \ldots\) |
| \(0n^2 + 0n + c = 9\) or \(c = 9\) | M1 | |
| their \(3 + b +\) their \(9 = 10\) or \(b = -2\) | M1dep | oe dep on M2 |
| \(3n^2 - 2n + 9\) | A1 | SC1 30 and 49 as the next two terms |
Additional guidance
In all cases \(a\), \(b\) and \(c\) refer to the general expression for the \(n\)th term of a quadratic sequence \(an^2 + bn + c\)
Condone \(n = 3n^2 - 2n + 9\) and accept any letter for \(n\)
Note that \(b = -2\) does not imply a specific number of marks