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Math Help - Recursive formula find closed something

  1. #1
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    Recursive formula find closed something

    Hi!
    There is this problem on recursive formulas I do not understand.
    The problem is:
    examine the recursive: a_{n+1}=\frac{a_n}{1+a_n} where a_0 =\frac{1}{2} and n=0,1,2,3,4...
    Find a closed formula of the type a_n = f(n) where f(n) is a expression that depends on n but not on for example a_{n-1}

    I can probably do it if I would understand what they want, closed formula????
    No idéa...where to start..
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  2. #2
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    Quote Originally Posted by Henryt999 View Post
    Hi!
    There is this problem on recursive formulas I do not understand.
    The problem is:
    examine the recursive: a_{n+1}=\frac{a_n}{1+a_n} where a_0 =\frac{1}{2} and n=0,1,2,3,4...
    Find a closed formula of the type a_n = f(n) where f(n) is a expression that depends on n but not on for example a_{n-1}

    I can probably do it if I would understand what they want, closed formula????
    No idéa...where to start..
    they want you to prove that a_n=\frac{1}{n+2}.
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  3. #3
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    Quote Originally Posted by Henryt999 View Post
    Hi!
    There is this problem on recursive formulas I do not understand.
    The problem is:
    examine the recursive: a_{n+1}=\frac{a_n}{1+a_n} where a_0 =\frac{1}{2} and n=0,1,2,3,4...
    Find a closed formula of the type a_n = f(n) where f(n) is a expression that depends on n but not on for example a_{n-1}

    I can probably do it if I would understand what they want, closed formula????
    No idéa...where to start..
    This problem tells you what a "closed formula" is! It "is an expression that depends on n" only. In other words, what you have always thought of, before, as a formula.

    If a_0= \frac{1}{2}, then a_1= \frac{\frac{1}{2}}{1+ \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{3}{2}}= \frac{1}{2}\frac{2}{3}= \frac{1}{3}.

    Then a_2= \frac{\frac{1}{3}}{1+ \frac{1}{3}}= \frac{\frac{1}{3}}{\frac{4}{3}} = \frac{1}{3}\frac{4}{3}= \frac{1}{4}.

    Now do you see where NonCommAlg got his formula?

    Of course you will need to prove that is correct. I recommend proving it by induction. Recursive formulas are ideally suited for induction.
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