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Discrete Math Question ( not really homework) about strong induction

  1. Mar 20, 2012 #1
    For some questions strong induction would test for cases n+1, n+2 and for other n+1,n+2,n+3, or other ways, why is that? Here's two examples


    Suppose that the only paper money consists of 3-dollar bills and 10-dollar bills. Find what dollar amounts
    can be made from a combination of these bills. Prove your result.
    SOLUTION: Clearly the amounts $3, $6, $9, $10, $12, $13, $15, $16, $18 can all be made using 3-dollar bills and 10-dollar bills. In addition any amount greater than $17 be made as well. To prove this we use strong induction.
    Base case: $18 can be made from 6 3-dollar bills, $19 can be made from one 10-dollar bill and three 3-dollar
    bills. And of course, $20 can be made from two 10-dollar bills.

    and the other example,

    prove that every amount of postage of 12 cents or more can be formed using just 4-cent and

    base case

    we can form postage of 12, 13 ,14 ,15 cents using three 4-cent stamps, two 4- cents stamps ... etc this shows that P(12),P(13), P(14) and P(15) are true.
     
  2. jcsd
  3. Mar 20, 2012 #2

    HallsofIvy

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    It is never necessary to "test for n+1, n+2, ...". You need to show a statement is true for n= 0, then show "if the statement is true for all [itex]k\le n[/itex] then it is true for n= k+1".

    (The difference between "strong induction" and "weak induction" is that strong induction allows you to assume "true for all [itex]k\le n[/itex]" while weak induction uses only "true for k= n".)
     
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