June 2016 Paper 2 Q12
12 Here are the first four terms of linear sequences \(X\) and \(Y\) and quadratic sequence \(Z\).
| Sequence \(X\) | 7 | 9 | 11 | 13 | .... |
|---|---|---|---|---|---|
| Sequence \(Y\) | 2 | 5 | 8 | 11 | .... |
| Sequence \(Z\) | 14 | 45 | 88 | 143 | .... |
(a) Work out the \(n\)th term of sequence \(X\). [2 marks]
(b) The \(n\)th term of sequence \(Y\) is \(\ 3n - 1\)
Using your answer to part (a), or otherwise, work out the \(n\)th term of sequence \(Z\).
Give your answer in the form \(\ an^2 + bn + c \ \) where \(a\), \(b\) and \(c\) are integers. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(2n + 5\) | B2 | B1 \(2n + k \quad k \ne 5\) |
Additional guidance
| Allow any letter eg \(2x + 5\) | B2 |
| Allow \(n =\) eg \(n = 2n + 5\) or \(n = 2x + 5\) | B2 |
| \(2n\) | B1 |
| \(2n + 5 = 0\) is B2 unless also seen with answer of \(-2.5\) which then scores B1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \((3n - 1)\ (\times)\) their \((2n + 5)\) | M1 | ft their (a) Brackets needed but may be recovered |
| \(6n^2 + 15n - 2n - 5\) | M1dep | 4 terms with at least 3 correct Implied by \(an^2 + 13n - 5 \ \ (a \ne 0)\) or \(6n^2 + 13n + b \ \ (b \ne 0)\) ft their two term linear expression in (a) |
| \(6n^2 + 13n - 5\) | A1ft | ft their two term linear expression in (a) |
| Alternative method 2 | ||
| (Second differences =) 12 or \(6n^2\) or \(a = 6\) | M1 | Seen at least once and not contradicted |
| \(14 - 6 \quad 45 - 24 \quad 88 - 54 \quad (143 - 96)\) or \(8 \qquad\quad 21 \qquad\quad 34 \qquad\quad (47)\) | M1dep | Subtracts \(6n^2\) from the terms in sequence \(Z\) |
| \(6n^2 + 13n - 5\) | A1 | |
| Alternative method 3 | ||
| Any two of \(a + b + c = 14\) \(4a + 2b + c = 45\) \(9a + 3b + c = 88\) \(16a + 4b + c = 143\) | M1 | |
| \(3a + b = 45 - 14\) and \(5a + b = 88 - 45\) | M1dep | oe Obtains two correct equations in same two variables from their equations |
| \(6n^2 + 13n - 5\) | A1 | |
| Alternative method 4 | ||
| (Second differences =) 12 or \(6n^2\) or \(a = 6\) | M1 | Seen at least once and not contradicted |
| \(3a + b = 45 - 14\) and substitutes \(a = 6\) | M1dep | oe eg1 \(5a + b = 88 - 45\) and substitutes \(a = 6\) eg2 \(7a + b = 143 - 88\) and substitutes \(a = 6\) |
| \(6n^2 + 13n - 5\) | A1 | |
| Alternative method 5 | ||
| (Second differences =) 12 | M1 | Seen at least once and not contradicted |
| \(14 + (45 - 14)(n - 1)\ +\) \(0.5 \times 12(n - 1)(n - 2)\) | M1dep | Using \(p + q(n - 1) + 0.5r(n - 1)(n - 2)\) \(p\) is 1st term \(q\) is 2nd term − 1st term \(r\) is second differences |
| \(6n^2 + 13n - 5\) | A1 | |
Additional guidance
| Allow any letter or mixed letters eg \(6x^2 + 13x - 5\) or \(6n^2 + 13x - 5\) | M1 M1 A1 |
| Allow \(n =\) eg \(n = 6n^2 + 13x - 5\) | M1 M1 A1 |
| \(6n^2 + 13n - 5 = 0\) is M1 M1 A1 unless also seen with solutions which then scores M1 M1 A0 |