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Math Help - question about induction

  1. #1
    Member Jones's Avatar
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    question about induction

    Hi,

    I had this induction problem which i couldn't figure out how to solve.

    for all n \geq 1 show that:  (1+x)^n \geq 1+nx

    The solution looked like this:
    \begin{array}{lcr}(1+x)^{k+1} = (1+x)(1+x)^k \\ <br />
\geq ~(1+x){\color{red}(1+kx)} \\<br />
=~1+kx+x+kx^2\\<br />
\geq~1+kx+x\\<br />
=~1+(k+1)x \end{array}

    \therefore (1+x)^{k+1} \geq 1+(k+1)x
    How can the stuff marked in red be valid?
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  2. #2
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    That's precisely the induction step -- you assumed that (1+x)^n \geq 1 + nx, so you get (1+x)(1+x)^k \geq (1+x)(1 + kx)

    And the rest follows easily..
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  3. #3
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    Quote Originally Posted by Jones View Post
    Hi,

    I had this induction problem which i couldn't figure out how to solve.

    for all n \geq 1 show that:  (1+x)^n \geq 1+nx

    The solution looked like this:
    \begin{array}{lcr}(1+x)^{k+1} = (1+x)(1+x)^k \\ <br />
\geq ~(1+x){\color{red}(1+kx)} \\<br />
=~1+kx+x+kx^2\\<br />
\geq~1+kx+x\\<br />
=~1+(k+1)x \end{array}

    \therefore (1+x)^{k+1} \geq 1+(k+1)x
    How can the stuff marked in red be valid?
    This is a rather well known thing called Bernoulli's Inequality, and it MUST be that x>-1

    Tonio
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  4. #4
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    Quote Originally Posted by Jones View Post
    Hi,

    I had this induction problem which i couldn't figure out how to solve.

    for all n \geq 1 show that:  (1+x)^n \geq 1+nx

    The solution looked like this:
    \begin{array}{lcr}(1+x)^{k+1} = (1+x)(1+x)^k \\ <br />
\geq ~(1+x){\color{red}(1+kx)} \\<br />
=~1+kx+x+kx^2\\<br />
\geq~1+kx+x\\<br />
=~1+(k+1)x \end{array}

    \therefore (1+x)^{k+1} \geq 1+(k+1)x
    How can the stuff marked in red be valid?

    at that part you bolded they are replacing the (1+x)^k with what we know.

    we know that  (1+x)^n \geq 1+nx.


    at the part you bolded, (1+x)(1+x)^k at this part we replace (1+x)^k with  (1+kx)  .

    leaving us with (1+x)(1+kx).
    Last edited by p00ndawg; October 19th 2009 at 03:52 PM.
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