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Math Help - prove by induction

  1. #1
    Junior Member Singular's Avatar
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    prove by induction

    Prove that if n>=12 then n can be written as a sum of 4's and 5's
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Singular View Post
    Prove that if n>=12 then n can be written as a sum of 4's and 5's
    See this thread for the general idea.

    Though in this case you will be assing 4 4's and taking away 3 5's. Take
    a bit of care to make sure that you have enough 5s to take 3 away.

    So suppose you can make n as a sum of 4's and 5's, add 4 4's, then if there
    are three or more 5's take 3 5's out, you now have a total of n+1.

    If there are no 5's take 5 4's (at this point we have a minimum of 7 of them)and replace with 4 5's, So now the 3 5's can be removed leaving a total of
    n+1.

    RonL
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  3. #3
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    Hello, Singular!

    Prove that if n > 12, then n can be written as a sum of 4's and 5's

    I haven't devised an inductive proof, but more of an intuitive one.


    Suppose n is a multiple of 4: . n = 4a, for a > 3.
    . . (Our number has at least three 4's.)

    We can make n + 1 by adding a 5 and removing a 4.

    We can make n + 2 by adding another 5 and removing another 4.

    We can make n + 3 by adding another 5 and removing another 4.


    The next number is n + 4, which is the next multiple of 4.
    . . And we can repeat the procedure.


    Therefore, we can construct any number n > 12.

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