Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=1$: $1^2-1+41=41$ is odd, thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$: we have $k^2-k+41$ is odd,
Step 3. Prove it is true for $n=k+1$: $(k+1)^2-(k+1)+41=k^2+2k+1-k-1+41=(k^2-k+41)+2k$. As $k^2-k+41$ is odd and $2k$ is even, their sum would be odd.
Thus, the statement is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
