I have a bunch of 3 and 5 dollar bills. prove that I can pay any amount(whole number)  with change (as long as amount  8 or larger). Explain simply,using proof by induction, and use symbols/algebra

### Problem Statement:
We are given $3 and $5 bills, and we need to prove that any amount $ n \geq 8 $ can be paid using some combination of these bills.

### Proof by Induction:

1. **Base Case**:
   We need to check that the smallest value, $ n = 8 $, can be made using just $3 and $5 bills.

   \[
   8 = 1 \times 3 + 1 \times 5
   \]
   So, the base case holds true.

2. **Inductive Hypothesis**:
   Assume that for some $ k \geq 8 $, we can form the amount $ k $ using $3 and $5 bills. That is, there exist non-negative integers $ x $ and $ y $ such that:
   \[
   k = 3x + 5y
   \]
   (This is our inductive assumption.)

3. **Inductive Step**:
   We must now prove that the amount $ k + 1 $ can also be formed using $3 and $5 bills.

   Since we know $ k = 3x + 5y $ by the inductive hypothesis, consider the following cases to construct $ k + 1 $:

   - If $ x \geq 1 $ (i.e., if there is at least one $3 bill), we can replace one $3 bill with two $5 bills to increase the amount by 1:
     \[
     k + 1 = 3(x - 1) + 5(y + 1)
     \]
     Thus, $ k + 1 $ can be made.

   - If $ x = 0 $, then $ k $ must be formed entirely by $5 bills. However, in this situation, $ k \geq 10 $ (since the smallest multiple of 5 is 5, and we’re assuming $ k \geq 8 $). So, it is always possible to subtract one $5 bill and add two $3 bills to get $ k + 1 $:
     \[
     k + 1 = 3 \times 2 + 5 \times (y - 1)
     \]
     This covers the case when $ x = 0 $.

Therefore, by both cases, we can form $ k + 1 $ from $3 and $5 bills.

### Conclusion:
By induction, we have shown that any amount $ n \geq 8 $ can be formed using $3 and $5 bills.

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Do you want more details or have any questions?

Here are five relative questions for deeper understanding:

1. What are other common examples of problems solved using induction?
2. How does this proof apply to other sets of bill denominations, like 4 and 7 dollars?
3. Why is the base case crucial for the proof by induction?
4. Can the inductive step be simplified or altered for different initial conditions?
5. How does the concept of modular arithmetic relate to this proof?

**Tip:** Always ensure that both the base case and inductive step are well-justified in an inductive proof.

I have a bunch of 3 and 5 dollar bills. Prove that I can pay any amount (whole number) with change (as long as the amount is 8 or larger). Explain simply, using proof by induction, and use symbols/algebra.

This problem demonstrates how to prove that any whole number amount (n ≥ 8) can be paid using 3 and 5 dollar bills, using proof by induction. Learn how to apply algebraic concepts, including the inductive hypothesis and step, to form combinations of these bills. Suitable for high school students in Grades 10-12.

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