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:
- Base Case: Prove that the statement is true for the initial case (usually the smallest value that meets the condition).
- Inductive Step: Assume that the statement is true for some arbitrary case , and then prove that it is true for .
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 (where ), the amount can be made using some combination of 3-dollar and 5-dollar bills.
- Inductive Step: Then prove that the amount 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:
- How would you rewrite this reasoning to explicitly include an inductive step?
- Can you identify other base cases that might be necessary for different starting amounts?
- What are the differences between this reasoning and a complete proof by induction?
- How would you apply proof by induction to a different problem, like the sum of an arithmetic series?
- 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.
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Math Problem Analysis
Mathematical Concepts
Proof by Induction
Combinatorics
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12
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