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 $|A_{n+1}| = |A_n^5 - 3| = |A_{n}|^5 + 3 > |A_n|^5$. If you can show that $|A_{n}|^5 > |A_n|$ you are done.

 

[Keru]

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

 

[CellCoree]

I understand your frustration with trying to mathematically prove the monotony of a recurrence relation. The best method to determine if a recurrence relation is increasing or decreasing is to use the first and second derivative tests.

In this particular case, we can use the first derivative test to show that the relation is decreasing. Taking the first derivative of the recurrence relation, we get:

An' = 5(An-1)^4 * An-1'

Since An-1' is always negative (as every term for n>=2 is a negative number raised to an odd power), and 5(An-1)^4 is always positive, An' will always be negative. This means that the value of An decreases as n increases.

Alternatively, we can also use the second derivative test by taking the second derivative of the recurrence relation. If the second derivative is negative, then the relation is decreasing. In this case, taking the second derivative gives us:

An'' = 20(An-1)^3 * (An-1')^2 - 20(An-1)^4 * An-1''

Again, since An-1' is always negative and (An-1)^3 and (An-1)^4 are always positive, An'' will always be negative. This confirms that the relation is decreasing.

I hope this helps you understand how to mathematically demonstrate the monotony of a recurrence relation. Keep in mind that these tests may not always work for every recurrence relation, but they are a good starting point. Good luck with your studies!