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Math Help - Induction help

  1. #1
    Junior Member
    Joined
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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.

    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) k + 1 <= 3^k +1
    is the equation 3^k + 3^k a simplification of the RHS of the above equation (1)?
    how do we get 3^k + 3^k from 3^k +1 ??

    thanks for any help.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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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.

    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) k + 1 <= 3^k +1
    is the equation 3^k + 3^k a simplification of the RHS of the above equation (1)?
    how do we get 3^k + 3^k from 3^k +1 ??

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

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

    Grandad
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