Lets say I am given " find the set" $$\bigcup_{k∈N}B_k$$ $$ B_k = \left[ \frac{3}{k}, \frac{5k+2}{k} \right) \cup \{10+k\} $$
I understand that $k$ is an argument and $N$ is a set however I have trouble understanding this problem
I think the answer is $$ \left\{ \left[\frac{3}{1}, \frac{5+2}{1} \right) \cup \{10+1\} \right\} \cup \cdots \cup \left\{ \left[\frac{3}{k}, \frac{5k+2}{k} \right) \cup \{10+k\} \right\} $$
To clarify your doubt: the second part (curly braces) is just a union of one element (just a natural number $10+k$ here) with the first part (the set). Let's look at those part by part.
For the first part note that we have a union of $[3/k,5+2/k)$. For a natural number k, 3/k runs from 0 to 3 and 2/k runs from 0 to 2. Hence, the union $\cup_{k\in N} [3/k,5+2/k)$ gives us (0,7).
The second part is easy. Just a set of natural numbers starting from 11 (10+1). The union $\cup_{k\in N} \{10+k\}$ gives us $\mathbb{N}-\{1,2,...,9,10\}$
So, the final answer is $$(0,7)\cup\mathbb{N}-\{7,8,9,10\}$$