ILL 1986 Problem 16

Given a positive integer $k$ , find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties: $$|S_j|=k \text{ for } j=1, \cdots , 5 , |\bigcup_{j=1}^{5} S_j | = n_k ;$$ $$|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \text{for } i=1,\cdots ,4$$