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Math Help - induction inequalities P1

  1. #1
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    induction inequalities P1

    Prove using the principle of mathematical induction that

    3^n > 4n^2 +5n for n=4,5...

    thank you!
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  2. #2
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  3. #3
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    Quote Originally Posted by christina View Post
    Prove using the principle of mathematical induction that

    3^n > 4n^2 +5n for n=4,5...

    thank you!
    This is very straightforward.

    3^n>4n^2+5n ?

    If this is true, we try to use it to prove if

    3^{n+1}>4(n+1)^2+5(n+1)

    (3)3^n>4(n^2+2n+1)+5(n+1) ?

    3^n+(2)3^n>4n^2+4(2n)+4+5n+5 ?

    3^n+[(2)3^n]>4n^2+5n+[4(2n)+9]

    Therefore, we only need ask if

    2(3^n)>4(2n)+9,\ n\ \ge\ 4

    3^n+3^n>4n+4.5+4n+4.5

    As n^2>n and 5n>4.5, for n=4 and above,

    only the first term needs to be tested.
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