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Math Help - Mathematical Induction

  1. #1
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    Mathematical Induction

    Prove by induction where n is a positive integer:

    The greatest number of regions that n straight lines can divide a circle is  \frac {1}{2} (n^2+n+2) , n\geq1

    The problem is, I do not know where/how to start. Can someone give me some hints to begin?
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  2. #2
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    Always start by proving that it starts.

    Try n = 1 and see.

    Try n = 2 and convince yourself. Is it really 4?

    Try n = 3 and become a disciple. Is it really 7?

    Are we really proving the premise or just the equation?
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  3. #3
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    Show for n=1: \frac{1^{2}+1+2}{2}=2........true.

    Assume P_{k} is true. This the induction hypothesis is

    R(k)=\frac{k^{2}+k+2}{2}

    We must show P_{k+1} is true.

    If we add a (k+1)st line, the regions increase by k+1.


    \frac{k^{2}+k+2}{2}+(k+1)=\frac{k^{2}+3k+4}{2}

    R(k+1)=\frac{(k+1)^{2}+(k+1)+2}{2}=\frac{k^{2}+3k+  4}{2}

    This shows that P_{k+1} is true and the induction holds. QED.
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  4. #4
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    Thanks. So we are proving just proving the equation.
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