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Math Help - Proof By Mathematical Induction - Example Help

  1. #1
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    Proof By Mathematical Induction - Example Help

    hello just started school 2 days ago. and we were told to copy down some examples and there was one were i think there is a mistake but not sure.

    Use mathematical induction to prove
    3^n >= 2n + 5 for all integers

    i understand all the steps untill it does this.

    3^(k+1)  >=  2(2k+5)
    3^(k+1)  >=  6+15
    3^(k+1)  >=  2k+7 <This line i do not understand.
    Therefore
    3^(k+1)  >=  2(k+1) + 5 <This line i do not understand.

    could some one prove this is true or false..
    Thanks in advance.
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  2. #2
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    Quote Originally Posted by name101 View Post
    hello just started school 2 days ago. and we were told to copy down some examples and there was one were i think there is a mistake but not sure.

    Use mathematical induction to prove
    3^n >= 2n + 5 for all integers

    i understand all the steps untill it does this.

    3^(k+1)  >=  2(2k+5)
    3^(k+1)  >=  6+15
    3^(k+1)  >=  2k+7 <This line i do not understand.
    Therefore
    3^(k+1)  >=  2(k+1) + 5 <This line i do not understand.

    could some one prove this is true or false..
    Thanks in advance.
    There are two lines I don't understand here, and they aren't the ones you noted. I'm kinda lost on the first two.

    Let me run through this one.

    I presume you've already done the base case so I won't redo that.

    We are assuming that, for some value of k:
    3^k \geq 2k + 5

    We need to show that this is true for the k + 1 case. So consider
    3^{k + 1} = 3 \cdot 3^k

    But we know that
    3^k \geq 2k + 5

    So
    3^{k + 1} = 3 \cdot 3^k \geq 3(2k + 5)<-- (1)

    Thus
    3^{k + 1} \geq 6k + 15 <-- (2)

    We also know that
    6k + 15 > 2(k + 1) + 5 <-- This step is easy to show. I leave it to you.

    Thus
    3^{k + 1} \geq 2(k + 1) + 5

    The lines I have labeled as (1) and (2) are what I think your lines 1 and 2 are meant to have been.

    -Dan
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    Hello, name101!

    Use mathematical induction to prove: . 3^n \:\geq \:2n + 5 for all integers
    This is not true for all integers.
    Could it have said: .for all n \geq 2 ?

    Here's the way I would do the proof.
    . . I suspect that "they" had the same idea.


    Verify the base case. . S(2)\!:\;\;3^2 \,\geq\,2\!\cdot2 + 5 . . . True!


    Assume S(k)\!:\;\;3^k \:\geq\:2k + 5


    Multiply both side by 3:. 3\cdot3^k \:\geq \:3(2k+5)

    . . and we have: . 3^{k+1}\:\geq\:6k + 15


    Now we note that: . 6k + 15 \;=\;2k + 4k + 2 + 5 + 8 \;=\;2k + 2 + 5 + 4k + 8

    . . . . . . . . . . . . =\:2(k+1) + 5 + (4k+8) \;>\;2(k+1) + 5 .
    . . . since k is positive


    Therefore, we have: . 3^{k+1} \:\geq \:2(k+1) + 5
    . . and the inductive proof is complete.

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  4. #4
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    Quote Originally Posted by Soroban View Post
    [size=3]This is not true for all integers.
    Could it have said: .for all n \geq 2 ?
    Ahhh yes I should of mentioned that fact.

    Quote Originally Posted by Soroban View Post
    Therefore, we have: . 3^{k+1} \:\geq \:2(k+1) + 5
    . . and the inductive proof is complete.

    Ahh thank you both. there were a couple of steps that were missing in the book.(I dont think it was a very good example to be learning off) but as i said.

    Thankyou
    and ill be back with more questions

    ~Regards
    Name101
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by name101 View Post
    (I dont think it was a very good example to be learning off)
    I think it is an excellent example to be learning from, given the nature of the substitutions that need to be made to solve the problem. That and the fact that there are many ways to go about doing it.

    -Dan
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  6. #6
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    Quote Originally Posted by topsquark View Post
    I think it is an excellent example to be learning from, given the nature of the substitutions that need to be made to solve the problem. That and the fact that there are many ways to go about doing it.

    -Dan
    I have been doing some more questions on the topic. and I revoke that statement.
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