1. **Problem Statement:** Use inductive reasoning to make a conjecture about the addition of an even integer and an odd integer, then prove it deductively. 2. **Inductive Reasoning:** We test several examples of adding an even integer and an odd integer. - Example 1: $2 + 3 = 5$ - Example 2: $6 + 3 = 9$ - Example 3: $4 + 3 = 7$ 3. **Observation:** In all cases, the sum is an odd number. 4. **Conjecture:** The sum of an even integer and an odd integer is always an odd integer. 5. **Deductive Proof:** - Let the even integer be $2n$ where $n$ is an integer. - Let the odd integer be $2m + 1$ where $m$ is an integer. - Their sum is: $$2n + (2m + 1) = 2(n + m) + 1$$ - Since $n + m$ is an integer, call it $k$, so the sum is: $$2k + 1$$ - This is the general form of an odd integer. 6. **Conclusion:** The sum of an even integer and an odd integer is always odd, confirming the conjecture.