Question 5 - A-Level Maths

Exam Board	OCR
Module	Further Additional Pure (Further Additional Pure)
Year	2018
Session	March
Topic	Sequences and Series

(a) Solve the recurrence relation $$X _ { n + 2 } = 1.3 X _ { n + 1 } + 0.3 X _ { n } \text { for } n \geqslant 0$$ given that $X _ { 0 } = 12$ and $X _ { 1 } = 1$.
(b) Show that the sequence $\left\{ X _ { n } \right\}$ approaches a geometric sequence as $n$ increases. The recurrence relation in part (i) models the projected annual profit for an investment company, so that $X _ { n }$ represents the profit (in $\pounds$ ) at the end of year $n$.
(a) Determine the number of years taken for the projected profit to exceed one million pounds.
(b) Compare your answer to part (ii)(a) with the corresponding figure given by the geometric sequence of part (i)(b).
(a) In a modified model, any non-integer values obtained are rounded down to the nearest integer at each step of the process. Write down the recurrence relation for this model.
(b) Write down the recurrence relation for the model in which any non-integer values obtained are rounded up to the nearest integer at each step of the process.
(c) Describe a situation that might arise in the implementation of part (iii)(b) that would result in an incorrect value for the next $X _ { n }$ in the process.