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Thread: Prove

  1. #1
    Member
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    Prove

    Hi,

    I'm stuck on the following problem:

    Let $\displaystyle h \ \epsilon \ [0,1]$
    Show that for every $\displaystyle n \ \epsilon \ N$

    $\displaystyle (1 + h)^n \leq 1 + (2^n -1)h$

    is true..


    I tried with math induction:

    $\displaystyle (1) \ \ \ n = 1$
    $\displaystyle 1 + h \leq 1 + (2 - 1)h$
    $\displaystyle 1 + h \leq 1 + h \ \ \ \surd $

    $\displaystyle (2) \ \ \ n \Rightarrow n + 1$

    $\displaystyle (1 + h)^{h+1} \leq 1 + (2^{n+1} - 1)h$

    $\displaystyle (1+h) (1+h)^n \leq 1 + (2 \cdot 2^n - 1)h$

    $\displaystyle (1+h) (1+h)^n \leq 1 + 2h \cdot 2^n - h$

    $\displaystyle (1+h) (1+h)^n \leq 1 - h + 2(h \cdot 2^{n-1})$

    now i'm lost. don't know how to continue.

    By the way, how do I write the "is element of" symbol? did i use the correct one above?
    and is there any way i could skip the [tex][/ math] in every line?
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  2. #2
    Junior Member utopiaNow's Avatar
    Joined
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    Quote Originally Posted by metlx View Post
    Hi,

    I'm stuck on the following problem:

    Let $\displaystyle h \ \epsilon \ [0,1]$
    Show that for every $\displaystyle n \ \epsilon \ N$

    $\displaystyle (1 + h)^n \leq 1 + (2^n -1)h$

    is true..


    I tried with math induction:

    $\displaystyle (1) \ \ \ n = 1$
    $\displaystyle 1 + h \leq 1 + (2 - 1)h$
    $\displaystyle 1 + h \leq 1 + h \ \ \ \surd $

    $\displaystyle (2) \ \ \ n \Rightarrow n + 1$

    $\displaystyle (1 + h)^{h+1} \leq 1 + (2^{n+1} - 1)h$

    $\displaystyle (1+h) (1+h)^n \leq 1 + (2 \cdot 2^n - 1)h$

    $\displaystyle (1+h) (1+h)^n \leq 1 + 2h \cdot 2^n - h$

    $\displaystyle (1+h) (1+h)^n \leq 1 - h + 2(h \cdot 2^{n-1})$

    now i'm lost. don't know how to continue.

    By the way, how do I write the "is element of" symbol? did i use the correct one above?
    and is there any way i could skip the [tex][/ math] in every line?
    So in the inductive step, we assume $\displaystyle (1 + h)^k \leq 1 + (2^k -1)h$ for some $\displaystyle k$.

    Then $\displaystyle (1 + h)^{k+1} = (1 + h)^k (1+h) \leq (1 + (2^k -1)h)(1+h)$ and $\displaystyle (1 + (2^k -1)h)(1+h) = 1 + 2^kh + (2^k - 1)h^2$

    Now $\displaystyle 1 + 2^kh + (2^k - 1)h^2 \leq 1 + 2^kh + (2^k - 1)h = 1 + 2^kh + 2^kh - h = 1 + (2^{k+1 } - 1)h$. Where this was true $\displaystyle (2^k - 1)h^2 \leq (2^k - 1)h$ since $\displaystyle h \in [0,1]$

    Therefore $\displaystyle (1+h)^{k+1} \leq 1 + (2^{k+1 } - 1)h$, as required.

    Oh and also, the element of symbol in latex is simply " \in ". So I used that above, you can click and see. And I don't know a shortcut to get around the constant math tags, sorry.
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