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Thread: prove using induction...

  1. #1
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    prove using induction...

    Prove by induction : show that $24 or more can be dispensed from an ATM using only $5 and $7 notes.

    how do i go about doing this?
    Last edited by mathman11; Jul 7th 2014 at 12:04 PM.
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    Re: prove using induction...

    Quote Originally Posted by mathman11 View Post
    Prove by induction : show that \$24 or more can be dispensed from an ATM using only \$5 and \$7 notes.

    how do i go about doing this?
    reposted fixing \$'s
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    Re: prove using induction...

    Quote Originally Posted by mathman11 View Post
    Prove by induction : show that \$24 or more can be dispensed from an ATM using only \$5 and \$7 notes.

    how do i go about doing this?
    1st can you do this with \$24?

    $2\times 5 + 2 \times 7 = 24$, so yes you can.

    Suppose you can do it with \$D. Show you can do it with \$(D+1).

    $D = 5m+7k,~~m,k \in \mathbb{N}$

    $D+1 = 5m + 7k + 1 = \dots$

    you should be able to finish from here.

    There are a couple different ways of obtaining D+1 and you'll need to use both of them.
    Last edited by romsek; Jul 7th 2014 at 12:37 PM.
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    Re: prove using induction...

    kool this helped!
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    Re: prove using induction...

    I would use five base cases:
    $\displaystyle 24 = 2\times 5 + 2\times 7$
    $\displaystyle 25 = 5\times 5+ 0 \times 7$
    $\displaystyle 26 = 1\times 5 + 3\times 7$
    $\displaystyle 27 = 4\times 5 + 1\times 7$
    $\displaystyle 28 = 0\times 5 + 4\times 7$

    For any $\displaystyle n\ge 24$, you can write $\displaystyle n=5k+24, n=5k+25, n=5k+26, n=5k+27,\text{ or }n=5k+28$ for some nonnegative integer $\displaystyle k$. It can only be written as one of them. Depending on how it can be written, you use the appropriate breakdown of the base number, then add $\displaystyle k$ $\$5$ notes.
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