# Monotony of a recurrence relation

1. Oct 21, 2014

### Keru

What method should i use to know if a recurrence relation is increasing or decreasing?
i was given the following relation:
A1 = 1
An=(An-1)^5 - 3

I know for sure it actually decreases since every term for n>=2 is a negative number raised to and odd number, but i don't know how to demonstrate it mathematically. I tried using induction, but it doesn't work...

Thanks to whoever can answer me.

2. Oct 21, 2014

### PeroK

Why do you think induction doesn't work?

It's clear that if $A_n < 0$ then $A_{n+1} < A_n$ and that's your inductive step.

3. Oct 21, 2014

### pasmith

You can show that $|A_n|$ is strictly increasing. If all terms are negative this shows that $A_n$ is strictly decreasing. For $n \geq 2$ you have that $|A_{n+1}| = |A_n^5 - 3| = |A_{n}|^5 + 3 > |A_n|^5$. If you can show that $|A_{n}|^5 > |A_n|$ you are done.

4. Oct 21, 2014

### Keru

Thanks guys, very quick and useful answers! I'll keep practising so i can see it by myself next time!