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Math Help - Mathematic induction Question 2

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    Mathematic induction Question 2

    Prove this by MI.
    ∑_(r=1)^3▒1/3^r = 1/2 - 1/(2(3^n))
    How do I display this properly? I write from MS words and paste here. I try wrapping with Math tag but it wont work.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by azuki View Post
    Prove this by MI.
    ∑_(r=1)^3▒1/3^r = 1/2 - 1/(2(3^n))
    We proceed by mathematical induction.

    Let P(n):~\sum_{r = 1}^n \frac 1{3^r} = \frac 12 - \frac 1{2 \cdot 3^n} for all n \in \mathbb{N},~n \ge 1

    P(1) is true, since \sum_{r = 1}^1 \frac 1{3^r} = \frac 13 = \frac 12 - \frac 1{2 \cdot 3}

    Assume P(n) is true, we now show that this implies P(n + 1) is true.

    Since P(n) is true, we have

    \sum_{r = 1}^n \frac 1{3^r} = \frac 12 - \frac 1{2 \cdot 3^r}

    Then

    P(n + 1):~ \sum_{r = 1}^{n + 1} \frac 1{3^r} = \sum_{r = 1}^n \frac 1{3^r} + \frac 1{3^{n + 1}}

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    i leave it to you to finish up. my last line was a crucial hint

    How do I display this properly? I write from MS words and paste here. I try wrapping with Math tag but it wont work.
    see here
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