1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mathematical induction/recusive sequences - just need a hint

  1. Sep 25, 2004 #1
    How do I prove by mathematical induction that the sequence given by

    [tex] a_{n+1}=3-\frac{1}{a_n} \qquad a_1=1 [/tex]

    is increasing?

    The difficulty in finding it myself is that recursive sequences are not familiar to me---i.e. usually, I am able do the following steps without a problem:

    (A) Show n=1 gives a true statement.
    (B) Assume it is true for n=k.
    (C) Get to n=k+1 and complete the proof.

    ... but this is not a regular sequence. Can anyone give me a tip?
  2. jcsd
  3. Sep 25, 2004 #2
    Not quite sure what kind of hint you're after... You wish to prove that a_(n + 1) > a_n for all n. The inductive step is basically showing that a_(k + 2) > a_(k + 1) if a_(k + 1) > a_k. If you work with the first inequality, you'll surely find something. Remember that you can relate a number in the sequence to its predecessor, e.g. a_(k + 2) = 3 - 1/a_(k + 1).
  4. Sep 25, 2004 #3


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    At n=1, we find a(2)=2>1=a(1), that is, the first step is verified.

    Assume a(k)>a(k-1)
    Prove that a(k+1)>a(k):

    a(k+1)-a(k)=3-1/a(k)-(3-1/a(k-1))=1/a(k-1)-1/a(k)>0, since a(k)>a(k-1)
  5. Sep 25, 2004 #4
    Oh... I've got it. Thanks. :smile:
  6. Sep 25, 2004 #5
    have you shown n=1?also, how does your prof let you play with the equations, does he/she let you pretty much play with them any way that is algebraically sensical or do you have to take a more controlled approach like playing with one side only, etc.?
  7. Sep 26, 2004 #6


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    What are you getting at?
    I've solely used the definitions of the term in the sequence; I haven't presupposed anything that should be proven :confused:
  8. Sep 28, 2004 #7
    to use induction, you show that it it true for n=1 before any other step is taken.
  9. Sep 28, 2004 #8

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    erm, and he did show n=1 is ok, in fact that was the first sentence in his proof.
  10. Sep 28, 2004 #9
    Sorry for the late reply---I didn't notice my notification was set to weekly.

    Anyhow, I have simply shown what I intended to prove to be valid for:
    Then, it would be true for all n.

    Well, The difficulty I had has already been resolved with your assistance, guys.

    Thank you very much indeed. :smile:
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Mathematical induction/recusive sequences - just need a hint
  1. Mathematical induction (Replies: 0)