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Proof by Induction- Sequences

  1. Mar 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Prove that an+2=an+1+an where a1=1 and a2=1 is monotonically increasing.


    2. Relevant equations
    A sequence is monotonically increasing if an+1≥an for all n[itex]\in[/itex]N.


    3. The attempt at a solution
    Base cases:
    a1≤a2 because 1=1.
    a2≤a3 because 1<2.

    Am I supposed to prove that an≤an+1 now? I'm not sure how to do that.
     
    Last edited: Mar 29, 2014
  2. jcsd
  3. Mar 29, 2014 #2

    SammyS

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    a2 ≥ a1 because 2 ≥ 1 . After all, a2 = 2 and a1 = 1 .

    Now, what you need to do:
    Assume that the statement is true for some k, where k ≥ 1 .
    I.e.:
    Assume that ak+1 ≥ ak .​
    From this, show that it follows that the statement is true for k+1.
    I.e.:
    Show that ak+2 ≥ ak+1 . ​
     
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