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Thread: Induction help

  1. #1
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    Induction help

    I understand the rules of prooving by induction, but I can't understand the math in this equation.
    If someone could explain what is happening at each step I would be very thankful.

    $\displaystyle k + 1 <= 3^k +1 <= 3^k + 3^k = 2.3^k <= 3.3^k = 3^k^+^1$

    The way I do math is down the page not across
    1)$\displaystyle k + 1 <= 3^k +1$
    is the equation $\displaystyle 3^k + 3^k$ a simplification of the RHS of the above equation (1)?
    how do we get $\displaystyle 3^k + 3^k$ from $\displaystyle 3^k +1$ ??

    thanks for any help.
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  2. #2
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    Hello dunsta
    Quote Originally Posted by dunsta View Post
    I understand the rules of prooving by induction, but I can't understand the math in this equation.
    If someone could explain what is happening at each step I would be very thankful.

    $\displaystyle k + 1 <= 3^k +1 <= 3^k + 3^k = 2.3^k <= 3.3^k = 3^k^+^1$

    The way I do math is down the page not across
    1)$\displaystyle k + 1 <= 3^k +1$
    is the equation $\displaystyle 3^k + 3^k$ a simplification of the RHS of the above equation (1)?
    how do we get $\displaystyle 3^k + 3^k$ from $\displaystyle 3^k +1$ ??

    thanks for any help.
    If $\displaystyle k\ge0$, then:
    $\displaystyle 3^k \ge 1$

    $\displaystyle \Rightarrow 1 \le 3^k$
    So if we add $\displaystyle 3^k$ to both sides of this inequality:
    $\displaystyle 3^k+1\le3^k+3^k$
    But
    $\displaystyle 3^k+3^k=2.3^k$
    So
    $\displaystyle 3^k+1\le2.3^k$
    And obviously
    $\displaystyle 2<3$
    So
    $\displaystyle 3^k+1\le3.3^k$
    And
    $\displaystyle 3.3^k=3^{k+1}$
    So
    $\displaystyle 3^k+1\le3^{k+1}$
    OK now?

    Grandad
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