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Monotony of a recurrence relation

  1. Oct 21, 2014 #1
    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. jcsd
  3. Oct 21, 2014 #2


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    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.
  4. Oct 21, 2014 #3


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    You can show that [itex]|A_n|[/itex] is strictly increasing. If all terms are negative this shows that [itex]A_n[/itex] is strictly decreasing. For [itex]n \geq 2[/itex] you have that [itex]|A_{n+1}| = |A_n^5 - 3| = |A_{n}|^5 + 3 > |A_n|^5[/itex]. If you can show that [itex]|A_{n}|^5 > |A_n|[/itex] you are done.
  5. Oct 21, 2014 #4
    Thanks guys, very quick and useful answers! I'll keep practising so i can see it by myself next time!
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