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Math Help - Show by mathematical induction:

  1. #1
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    Post Show by mathematical induction:

    We have:

    a1 = 2, a(n+1) = (an^2 +2) / 2an


    Then comes the induction proof:
    " Use mathematical induction to show that: an>= 0 for all n"

    This should also mean that a(n+1) is greater than, or equal to 0. So that:
    a(n+1) >= 0
    (an^2 +2) / 2an >= 0
    an^2 +2 >= 0

    Where do I go from here?
    Or could I just say that an has to be greater than 0, because if an=0 then the denominator for a(n+1) would be 0?
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  2. #2
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    Your original conditions are almost unreadable.

    Are you saying

    a_1 = 2 and a_{n + 1} = \frac{a_n^2 + 2}{2a_n}?
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  3. #3
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    Look at this outline of every induction proof. That post is about a different problem, but every induction proof must follow those steps.
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  4. #4
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    Yes Don't know how to write sub/sup text...
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  5. #5
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    I don't think induction is necessary. To show that this is always positive, show that the numerator and denominator are always the same sign (I can't see any reason why either of them would ever be negative...)
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  6. #6
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    I agree, but the task says I have to prove it by induction
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  7. #7
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    Strictly speaking, induction is necessary. The numerator of (a_n^2 +2) / 2a_n is always positive, so one has to know that a_n is positive. This can only come from the induction hypothesis.
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  8. #8
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    Maybe I could say:

    The only way that a(n+1) < 0, is if an < 0, since the numerator is always greater than 0 because an^2 > 0.
    Therefor all I have to prove is that a(n+1) always is greater than zero.

    So if:
    a(n+1) >= 0

    Then:
    (an^2 +2) / 2an >= 0
    an^2 +2 >= 0

    Which is always greater than zero.

    Does that sound correct?
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  9. #9
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    Unfortunately, no.
    all I have to prove is that a(n+1) always is greater than zero.
    This is equivalent to the original problem that asks to show that a(n) >= 0 for all n. So you have not simplified anything.

    So if:
    a(n+1) >= 0
    You can't assume what you have to prove. It's like expecting that the cashier in a grocery will give you cash for buying stuff.

    There is no way around the steps that I mentioned above. The only way to gloss over them is knowing them so well that it is boring to write every detail.
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  10. #10
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    Ok.

    Obviously the statement is true for n=1. But how do I proceed from here?
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  11. #11
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    Assume that a_{k + 1} is nonnegative.

    That would mean that \frac{a_k^2 + 2}{2a_k} \geq 0.


    We need to show that a_{k+ 2} is nonnegative.

    a_{k + 2} = \frac{a_{k + 1}^2 + 2}{2a_{k + 1}}

    Since a_{k + 1} \geq 0 that means 2a_{k + 1} \geq 0.

    And the numerator is always nonnegative.

    Since the numerator and denominator are both nonnegative, that means a_{k + 2} is nonnegative.

    Q.E.D.
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  12. #12
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    Always using n --> n+1, made me forget that n+1 --> n+2 is also allowed. Thanks
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  13. #13
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    Quote Originally Posted by jenkki View Post
    Always using n --> n+1, made me forget that n+1 --> n+2 is also allowed. Thanks
    You can use any two consecutive general terms. The point is to show by the base step that you knock over the first domino, and by the inductive step that knocking over any domino will knock over the next one.
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  14. #14
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    There is also a b) part to this task, which I can't seem to solve

    It goes:
    Show that a_{n}^2 -2 \geq 0 for all values of n


    This is what I'm thinking:
    If we assume that: a_{n}^2 \geq 2, we have to show that this is also the case for a_{n+1}^2 \geq 2


    From before we have that: a_1 = 2 and a_{n + 1} = \frac{a_n^2 + 2}{2a_n}. And we also know that: a_{n} \geq 0 for all n.


    I have tried putting a_{n+1}^2 = (\frac{a_n^2 + 2}{2a_n})^2.

    But from here I'm not able to show that a_{n+1}^2 \geq 2
    Any ideas?
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  15. #15
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    Quote Originally Posted by jenkki View Post
    There is also a b) part to this task, which I can't seem to solve

    It goes:
    Show that a_{n}^2 -2 \geq 0 for all values of n


    This is what I'm thinking:
    If we assume that: a_{n}^2 \geq 2, we have to show that this is also the case for a_{n+1}^2 \geq 2


    From before we have that: a_1 = 2 and a_{n + 1} = \frac{a_n^2 + 2}{2a_n}. And we also know that: a_{n} \geq 0 for all n.


    I have tried putting a_{n+1}^2 = (\frac{a_n^2 + 2}{2a_n})^2.

    But from here I'm not able to show that a_{n+1}^2 \geq 2
    Any ideas?
    P(k)

    \left(a_k\right)^2-2\ \ge\ 0


    P(k+1)

    \left(a_{k+1}\right)^2-2\ \ge\ 0


    Proof

    We are trying to show

    \displaystyle\left(\frac{\left(a_k\right)^2+2}{2a_  k}\right)^2-2\ \ge\ 0

    if P(k) is valid

    \displaystyle\frac{\left(a_k\right)^4+4\left(a_k\r  ight)^2+4}{4\left(a_k\right)^2}-2\ \ge\ 0 ?

    \displaystyle\frac{\left(a_k\right)^4+4\left(a_k\r  ight)^2+4}{4\left(a_k\right)^2}\ \ge\ 2 ?

    \left(a_k\right)^4+4\left(a_k\right)^2+4\ \ge\ 8\left(a_k\right)^2 ?

    \left(a_k\right)^4-4\left(a_k\right)^2+4\ \ge\ 0 ?

    \left(\left(a_k\right)^2-2\right)^2\ \ge\ 0 ?
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