AUSTRIAN-POLISH 1978 Problem 8

For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side.

(a) Show that the sequence converges, for every fixed integer $k\ge 1$ . (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.