June 2022 Paper 2 Q9
9 The first three terms of a linear sequence are
\[30 \qquad\quad 30 + 4k \qquad\quad 30 + 8k\]where \(k\) is a constant.
(a) Work out an expression, in terms of \(k\), for the 4th term.
Give your answer in its simplest form. [1 mark]
(b) The 100th term of the sequence is 525
Work out the value of \(k\). [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(30 + 12k\) or \(12k + 30\) | B1 | allow factorised eg \(6(5 + 2k)\) |
Additional guidance
| \(30 + 12k\) seen in working but incorrect answer eg \(5 + 2k\) or \(-2.5\) | B0 |
| Answer line \(30 + 12k\) and expression for the \(n\)th term eg \(30 + 4nk - 4k\) | B0 |
| \(30 + 8k + 4k\) | B0 |
| \(30 + 12k\) unambiguously indicated as 4th term (eg in given sequence) with answer line blank | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Works out a correct expression for the 100th term | ||
| \(30 + 99 \times 4k\) or \(30 + 396k\) or \(100 \times 4k + 30 - 4k\) | M1 | oe eg \(30 + (100 - 1) \times 4k\) or \(30 + 4k + 98 \times 4k\) or \(30 + 8k + 97 \times 4k\) or \(30 + 12k + 96 \times 4k\) |
| \(99 \times 4k = 525 - 30\) or \(396k = 495\) or \(495 \div 396\) | M1dep | oe terms must be collected in an equation eg \(396k - 495 = 0\) |
| 1.25 or \(\dfrac{5}{4}\) or \(1\dfrac{1}{4}\) | A1 | oe eg \(\dfrac{495}{396}\) |
| Alternative method 2 Uses a common difference (eg \(d\)) | ||
| \(30 + 99 \times d\) or \(30 + 99d\) | M1 | oe eg \(30 + (100 - 1) \times d\) |
| \(4k = \dfrac{525 - 30}{99}\) or \(4k = \dfrac{495}{99}\) or \(4k = 5\) or \(5 \div 4\) | M1dep | oe terms must be collected in an equation eg \(4k - 5 = 0\) |
| 1.25 or \(\dfrac{5}{4}\) or \(1\dfrac{1}{4}\) | A1 | oe eg \(\dfrac{495}{396}\) |
| Alternative method 3 Uses their (a) to work out an expression for the 100th term | ||
| their (a) \(+ 96 \times 4k\) or their (a) \(+ 384k\) | M1 | their (a) must be in terms of \(k\) their (a) cannot be \(30 + 4k\) or \(30 + 8k\) |
| Collection of terms for their (a) \(+ 384k = 525\) | M1dep | their (a) must be of the form \(c + dk\) \(c \neq 0 \quad d \neq 0\) |
| Solution to their equation rounded to 1 dp or better | A1ft | ft their (a) and M2 |
Additional guidance
| Ignore simplification or conversion if correct answer seen | |
| Alt 1 Do not allow M1 if seen embedded eg in formula for \(\mathrm{S}_n\) | |
| Alt 3 (a) \(12k \qquad\) (b) \(12k + 384k \qquad 396k = 525 \qquad 1.326\) | M1M0A0ft |
| Alt 3 (a) \(30 + 16k \qquad\) (b) \(30 + 16k + 384k \qquad 400k = 525 - 30 \qquad 1.238\) | M1M1A1ft |
| Alt 3 (a) \(12k + 60 \qquad\) (b) \(12k + 60 + 96 \times 4k \qquad 396k = 525 - 60 \qquad 1.2\) | M1M1A1ft |