Questions for first year students at the University.

Question

I will help teach in a introductory class in mathematics for engineers in applied math at the University. Anyone have any good and cool favorite questions or know where I can find some? Anything is welcome

For the moment there is some questions like

• write: $1 + 2 + 3 + ... + 100$ as a sum

• show that $XY-YX = I$ has no solution for square matrices.

• $p | (n^2 - n)$ ($p$ is prime).

Thanks

Answer 1

Ask them to prove that the product of any $k$ consecutive integers is a multiple of $k!$. And then just show them how powerful the formula for $\binom{n}{k}$ they learned in high school really is.

Answer 2

1. Ask them to prove that there are infinitely many primes. I always found that to be a beautiful result.

2. Prove that $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$. I guess engineers will be interested in methods to linearize equations.

Answer 3

Try asking students to solve for $x$ in the equation $x+\sin(x)=0$.

The solution $x=0$ seems trivial, but it can only be solved by numerical methods, a course of which engineering majors will benefit greatly from.