June 2024 Paper 1 Q4
4 A sequence has terms \(u_1, u_2, u_3, \ldots\) defined by \(u_1 = 2\) and \(u_{n+1} = 1 - \dfrac{1}{u_n}\) for \(n \geqslant 1\).
| Scheme | Marks | AO |
|---|---|---|
| \(u_2 = \tfrac{1}{2}\) | B1 | 1.1 |
| \(u_3 = -1,\ u_4 = 2\) | B1FT | 1.1 |
| [2] |
Notes
B1: Or 0.5
Must be seen as \(\tfrac{1}{2}\) and not just \(1 - \tfrac{1}{2}\)
B1FT: FT their \(u_2\)
Both as simplified numerical values
| Scheme | Marks | AO |
|---|---|---|
| Periodic, with period 3 | B1 | 1.2 |
| [1] |
Notes
B1: Any correct description, such as repeating.
Condone just ‘periodic’ without the period being stated.
ISW an incorrect period eg ‘periodic with period 4’
B0 if additional incorrect description eg ‘periodic AP’
Allow recurring, repetitive, cyclic etc
Condone looping, circling etc
Do not allow harmonic or alternating, even if with another correct description
B0 for divergent or oscillating, unless additional detail eg between 3 values
Must have a periodic sequence in (a) (with period of at least 3) to gain credit for description
See appendix for further examples (below).
Appendix: exemplar responses for Q4(b)
| Response | Mark | Comment |
|---|---|---|
| Periodic with order 4 | B1 isw | Ignore any attempt to give the period of the sequence as it is just the ‘general behaviour’ that is required. |
| Repeating, or repetitive or recurring | B1 | These are all acceptable descriptions. |
| Cyclic, or circling or looping | B1 | These are all acceptable descriptions. |
| Repetitive and infinite | B1 | The infinite isn’t incorrect, so can be ignored. |
| Oscillating sequence | B0 | MS states B0 for oscillating on its own. |
| Periodic oscillating | B1 isw | Ignore the comment oscillating if with an acceptable description. |
| Divergent | B0 | Insufficient description on its own. |
| Repeating and divergent | B1 isw | The divergent isn’t incorrect, so can be ignored. |
| Repeating and convergent | B0 | An incorrect description, alongside an acceptable statement cannot be condoned. |
| Harmonic | B0 | Incorrect description. |
| Periodic harmonic sequence | B0 | An incorrect description, alongside an acceptable statement cannot be condoned. |
| Scheme | Marks | AO |
|---|---|---|
| \(u_1 + u_2 + u_3 = 2 + 0.5 - 1 = 1.5\) so total goes up by 1.5 each time soi | M1 | 3.1a |
| \(73 = 70.5 + 2 + 0.5\) \(\phantom{73} = (47 \times 1.5) + 2 + 0.5\) | M1 | 1.1 |
| \(k = 143\) | A1 | 1.1 |
| [3] |
Notes
M1: Identify that every block of three terms will increase the total by 1.5 (allow use of \(2 + 0.5 - 1\) instead).
Can still award M1 if using the sum of three of their consecutive terms
Must have a periodic sequence in (a) to gain any credit for method (but condone one with a period other than 3)
M1: Attempt to identify the number of terms needed eg 47 blocks plus 2 more terms.
\(73 \div 1.5\) is sufficient for M1 M1
Allow M1 if using blocks of 1.5 to try to find a sum of 73 eg \(48 \times 1.5 = 72\)
Can still get M1 if attempting to use the sum of their three terms
A1: Obtain \(k = 143\)