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Math Help - prove by induction...alittle lost

  1. #1
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    Red face prove by induction...alittle lost

    hi,

    can anyone explain these steps to me? im abit lost? especially step 2 and 3

    Exercise 4(c) To prove that 3^n > n^2, i.e., 3^n − n^2 > 0, using proof
    by induction, we first check it for n = 1: i.e., 3^1 = 3 > 1^2 = 1 . X
    Inductive step: now assume it is true for n = k, i.e.,

    solution
    assume that
    3k − k^2 > 0. For n = k + 1

    3^k+1 − (k + 1)^2 = 3 3^k − (k + 1)^2

    = 3(3^k − k^2 + k^2) − (k + 1)^2 [2]

    = 3(3^k − k^2) + 3k^2 − k2 − 2k − 1 [3]

    = 3(3^k − k^2) + k^2 + k^2 − 2k − 1

    = 3(3^k − k^2) + k^2 + (k − 1)2 − 2
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  2. #2
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    Assume that k > 1\,\& \,3^k  > k^2 .
    Then 3^{k + 1}  = \left( 3 \right)3^k  > 3k^2  \ge k^2  + 2k + 1 = \left( {k + 1} \right)^2
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  3. #3
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    hey Plato,

    thanks for the reply,

    one thing on line [2] how does

    3x3^k become 3(3^k − k^2 + k^2) ??
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  4. #4
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    Quote Originally Posted by jonbob81 View Post
    3x3^k become 3(3^k + k^2 + k^2) ??
    k > 1\,\& \,2k^2  > 2k + 1
    3k^3  = k^3  + 2k^3  > k^3  + 2k^2  > k^2  + 2k + 1
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