Providing Tight Example in Approximation Algorithm Analysis
That depends on your definition of approximation ratio. Normally the approximation ratio is defined as the worst ratio between optimal solution and the one produced by your algorithm. If this is the case, all you need to show that the ratio is tight is come up with one bad example.

Sometimes, however, you prove something like $ALG \leq 2OPT + 1$. This means that your approximation ratio is really $2 + o(1)$. To show that this is tight, you will need an example for infinitely many sizes (but not necessarily for a generic size; perhaps all your examples have even size).