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Math Help - proof by induction

  1. #1
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    proof by induction

    Hello,

    Could someone help me prooving the following:

    If

     x+1 >= 0

    then

    (1+x)^n >= 1+nx

    for all natural numbers.

    I hope that i have written everything correctly (this is my first time).

    NB! How do i type in equations and stuff?

    Thank you very much.
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  2. #2
    o_O
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    Please no double posting. On typing math symbols, see this thread for more: LaTex Tutorial

    You can see the code that generates the image by clicking on it.

    As for your problem, let's look at the inductive step. Assuming that \color{red} (1+x)^k \geq 1 + kx, our goal is to prove that (1+x)^{k+1} \geq 1 + (k+1)x.

    Looking at the LHS, notice that: (1+x)^{k+1} = {\color{red}(1+x)^k}(1+x)

    Now I highlighted the part in red for a reason .. and it shouldn't be too hard to show that this is bigger than 1 + (k+1)x.
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  3. #3
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    Proof by induction

    Hello again,

    Sorry for the inconvenience. The thing is that what you mentioned before is exactly what I have already done . . . and then I get stuck. I can't seem to get the "great idea" (maybe because I havn't slept all night).

    What I'm trying to do is to reach the conslusion (instead of showing the truthfulness of an inequality):

     (1+x)^{n+1} \geq 1+(n+1)x

    Now from calculus I know that:

     (1+x)^{n+1}= (1+x)^{n}(1+x)

    Using the inductive step we may write:

     (1+x)^{n}(1+x) \geq (1+nx)(1+x)

    This is where I get stuck.

    Your help is greatly appreciated.

    Thanks.
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  4. #4
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    Quote Originally Posted by surjective View Post
    Hello again,

    Sorry for the inconvenience. The thing is that what you mentioned before is exactly what I have already done . . . and then I get stuck. I can't seem to get the "great idea" (maybe because I havn't slept all night).

    What I'm trying to do is to reach the conslusion (instead of showing the truthfulness of an inequality):

     (1+x)^{n+1} \geq 1+(n+1)x

    Now from calculus I know that:

     (1+x)^{n+1}= (1+x)^{n}(1+x)

    Using the inductive step we may write:

     (1+x)^{n}(1+x) \geq (1+nx)(1+x)

    This is where I get stuck.

    Your help is greatly appreciated.

    Thanks.
    Continuing on,

    if (1+x)^n\ \ge\ 1+nx, then the following must be true

    (1+x)^{n+1}\ \ge\ 1+(n+1)x

    Proof

    (1+x)^n(1+x)\ \ge\ (1+nx)(1+x) if the first statement is true

    (1+x)^n(1+x)\ \ge\ 1+x+nx+nx^2

    (1+x)^{n+1}\ \ge\ [1+(n+1)x]+nx^2

    This means that if (1+x)^n\ \ge\ 1+nx

    then (1+x)^{n+1}\ \ge\ [1+(n+1)x]+nx^2

    and therefore

    (1+x)^{n+1}\ \ge\ 1+(n+1)x definately.

    That's the inductive step.
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  5. #5
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    Proof by induction

    Hello,

    Thank you very much. It seems that I was only one step away from completing the proof. Thank you all very much.
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