Monotony of a recurrence relation

[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.

 

[PeroK]

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.

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.

 

[pasmith]

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.

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 &gt; |A_n|^5. If you can show that |A_{n}|^5 &gt; |A_n| you are done.

 

[Keru]

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