Edexcel A-Level Mathematics AS P2.4.3 Increasing Sequences Decreasing Sequences and Periodic Questions

The direct evidence is focused: show the sequence repeats, name the order, then split long sums into full cycles plus any remainder terms.

Exam points

Calculate successive terms such as u2, u3 and u4 to show the sequence repeats.
State the order of a periodic sequence once the repeating cycle is established.
Split a finite sum into complete cycles and remainder terms before evaluating it.

[Maximum number: 3]

(a)

A sequence $u_{1}, u_{2}, u_{3}, \ldots$ is defined by

\begin{aligned} u_{1} & =3 \\ u_{n+1} & =\frac{u_{n}-3}{u_{n}-2} \quad n \in \mathbb{N} \end{aligned}

[ 3 ]

(i)

Show that this sequence is periodic.

[ 2 ]

$\quad\left(u_{1}=3,\right) u_{2}=0, u_{3}=\frac{3}{2}, u_{4}=3, \ldots$ $\left(u_{1}=3\right), u_{2}=0, u_{3}=\frac{3}{2}, u_{4}=3, \ldots$ and e.g. states that $u_{4}=u_{1}$ so sequence will repeat (is
periodic)
(2)

(ii)

State the order of this sequence.

[ 1 ]

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Edexcel A-Level Mathematics AS P2.4.3 Increasing Sequences Decreasing Sequences and Periodic Questions question 1

Question (a)

Question (i)

Question (ii)