And so on - and on - and on
Can you find the value of this function involving algebraic fractions for x=2000?
Age
16 to 18
Challenge level
Problem
Let
Evaluate $f_{2000}(2000)$
Evaluate $f_{2000}(2000)$
Getting Started
The solution needs you to be systematic.
Start with $ f_{0} $, then work out $ f_{1}$, then work out $ f_{2}$
Student Solutions
A well presented solution from Richard of The Royal Hospital School reflected those of a number of other solvers including Kevin of Langley Grammar, Jeff from New Zealand and Andrei of Tudor Vianu School. Well done to all of you.
We are given that:
This implies that:
Creating functions of $x$ when $n = 1, 2$ and $3$ gives:
Therefore:
Teachers' Resources
It is worth emphasising the cyclic nature of this recurrence relation.
What happens with different starting numbers and why?