Challenging +1.8 This is a Further Maths question requiring students to compute several terms of a recursively-defined sequence to identify the period, then prove periodicity. While the computation is straightforward, it requires systematic calculation of 5-7 terms, pattern recognition, and a clear proof structure. The recursive formula involves division which is less routine than addition/multiplication recurrences, and proving periodicity (showing u_{n+5} = u_n) requires careful algebraic manipulation. This goes beyond standard A-level but is a structured problem with a clear path once the pattern is spotted.
1 The sequence \(\left\{ u _ { n } \right\}\) is defined by \(u _ { 0 } = 2 , u _ { 1 } = 5\) and \(u _ { n } = \frac { 1 + u _ { n - 1 } } { u _ { n - 2 } }\) for \(n \geqslant 2\).
Prove that the sequence is periodic with period 5.
Explanation that u = u and u = u ⇒ periodicity, period 5
Answer
Marks
0 5 1 6
1.1a
1.1
1.1
Answer
Marks
2.4
M1
M1
A1
E1
Answer
Marks
[4]
Use of the given r.r. (with at least u correct)
2
Repeated use of the given r.r. (with at leastu correct)
3
u , u & u correct
4 5 6
(since a second-order r.r.)
Answer
Marks
Guidance
1
1
11
Question 1:
1 | u =3
2
u =0.8
3
u =0.6, u =2 and u =5
4 5 6
Explanation that u = u and u = u ⇒ periodicity, period 5
0 5 1 6 | 1.1a
1.1
1.1
2.4 | M1
M1
A1
E1
[4] | Use of the given r.r. (with at least u correct)
2
Repeated use of the given r.r. (with at leastu correct)
3
u , u & u correct
4 5 6
(since a second-order r.r.)
1 | 1 | 11 | 41 | 51
1 The sequence $\left\{ u _ { n } \right\}$ is defined by $u _ { 0 } = 2 , u _ { 1 } = 5$ and $u _ { n } = \frac { 1 + u _ { n - 1 } } { u _ { n - 2 } }$ for $n \geqslant 2$.\\
Prove that the sequence is periodic with period 5.
\hfill \mbox{\textit{OCR Further Additional Pure 2019 Q1 [4]}}