Answer
We can form all amounts except \$1 and $3.
Work Step by Step
--We can form all amounts except \$1 and \$3.
-Let P(n) be the statement that we can form n dollars using just 2-dollar and 5-dollar bills.
- We want to prove that P(n) is true for all n ≥ 5. (It is clear that \$1 and \$3 cannot be formed and that \$2 and \$4 can be formed.)
For the basis step, note that 5 = 5 and 6 = 2+2+2. Assume the inductive hypothesis, that P(j) is true for all j with 5 ≤ j ≤ k, where k is an arbitrary integer greater than or equal to 6.
--- We want to show that P(k + 1) is true. Because k−1 ≥ 5, we know that P(k−1) is true, that is,
that we can form k −1 dollars. Add another 2-dollar bill, and we have formed k+1 dollars.
