Math Problem Statement

is this response using proof by induction "Alright, let's tackle this with a bit of algebra and logical reasoning. We want to show that any amount of money, let's call it M , greater than 8 dollars, can be made using just 3-dollar and 5-dollar bills.

First, let's establish a foundation with some specific examples to see the pattern:

For $8: We can't make $8 exactly with just 3s and 5s, so this is our starting point.

For $9: We can use three 3-dollar bills (3x3).

For $10: We can use two 5-dollar bills (2x5).

For $11: We can't make $11 exactly with just 3s and 5s.

For $12: We can use four 3-dollar bills (4x3).

Now, notice something interesting:

Once we have $12 (which is 3x4), we can make any amount above by adding more 3-dollar bills.

But what about the numbers we skipped, like $11? Well, we don't need to worry about them because we're looking for any amount greater than $8, and we've found a way to make $9, $10, and $12. From $12 onwards, we can add 3-dollar bills to cover all subsequent amounts.

To put this into a general form, we can express amounts using 3-dollar and 5-dollar bills as follows:

M = 3a + 5b

where M is the total amount, a is the number of 3-dollar bills, and b is the number of 5-dollar bills.

We've shown that for M = 9 and M = 10 , we have solutions. For any M > 10 , we can use the combinations we found for $9 and $10 and just add enough 3-dollar bills to reach M . Specifically, for any M > 10 , we can write:

M = 9 + 3n or M = 10 + 3n

for some non-negative integer n . Here, n represents the additional number of 3-dollar bills we can add to our base cases of $9 and $10 to reach any desired amount M .

This shows that using a combination of 3-dollar and 5-dollar bills, we can create any amount greater than $8."

Solution

The response you've provided does include some elements of a proof by induction, but it is not a complete formal proof by induction. Proof by induction typically involves two main steps:

  1. Base Case: Prove that the statement is true for the initial case (usually the smallest value that meets the condition).
  2. Inductive Step: Assume that the statement is true for some arbitrary case kk, and then prove that it is true for k+1k + 1.

Let’s analyze the response in light of these two steps:

Base Case:

The response establishes the base cases for specific amounts (e.g., $9, $10, and $12), showing that they can be constructed using combinations of 3-dollar and 5-dollar bills. The base case here can be seen as showing that amounts $9 and $10 can be made, which covers the smallest values greater than $8.

Inductive Step:

The inductive step is not explicitly mentioned, but it is somewhat implied when the response states:

  • "Once we have $12 (which is 3x4), we can make any amount above by adding more 3-dollar bills."

However, for a complete proof by induction, one would explicitly state:

  • Inductive Hypothesis: Assume that for some M=kM = k (where k12k \geq 12), the amount can be made using some combination of 3-dollar and 5-dollar bills.
  • Inductive Step: Then prove that the amount M=k+1M = k + 1 can also be made using the same method. This proof would involve showing that adding one more 3-dollar bill to an amount that can already be made (e.g., $k) results in the next amount (e.g., $k + 3), and thus all amounts greater than $12 can be generated.

Conclusion:

The response does a good job of reasoning about the problem and hinting at the inductive structure by showing how to construct amounts greater than $12. However, it stops short of explicitly framing this reasoning as a formal induction proof. To complete the proof by induction, you would need to explicitly state and verify the inductive step.

Would you like further details on formalizing this into a complete proof by induction?


5 Relative Questions:

  1. How would you rewrite this reasoning to explicitly include an inductive step?
  2. Can you identify other base cases that might be necessary for different starting amounts?
  3. What are the differences between this reasoning and a complete proof by induction?
  4. How would you apply proof by induction to a different problem, like the sum of an arithmetic series?
  5. Could there be another method, aside from induction, to prove this statement?

Tip:

When writing a proof by induction, always ensure the inductive hypothesis is clearly stated and that the inductive step logically follows from it, covering all potential cases.