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Math Help - Proving Inequality

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    Proving Inequality

    Prove that 1+2n <= 3^n by using mathematical induction.
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  2. #2
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    Hello UC151CPR
    Quote Originally Posted by UC151CPR View Post
    Prove that 1+2n <= 3^n by using mathematical induction.
    First, note that for n\ge0, 3^n\ge1

    \Rightarrow 2\cdot3^n\ge 2 (1)

    Then suppose that P(n) is the propositional function 1+2n\le 3^n

    Then P(n) \Rightarrow 1+2n + 2 \le 3^n+2

    \Rightarrow 1 +2(n+1) \le 3^n + 2\cdot3^n, using (1)

    \Rightarrow 1 +2(n+1) \le 3^n(1+2)=3^{n+1}

    So P(n) \Rightarrow P(n+1)

    P(1) is 1+2\le3^1, which is true.

    Hence by Induction, P(n) is true for all n \ge1

    Grandad
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