Final answer:
The order of increasing atomic radius for F, K+, Br-, and Mg, considering the periodic table trends, is Mg < F < K+ < Br-.
Step-by-step explanation:
To rank F, K+, Br-, and Mg in order of increasing atomic radius, we need to consider the trends in atomic size on the periodic table. Within a period, atomic radius tends to decrease from left to right due to increasing nuclear charge, causing the attraction between the nucleus and electrons to be stronger and pulling the electrons closer. Within a group, atomic radius increases as you move down, because each row down adds another electron shell.
Now, considering the elements and ions given, K+ has lost an electron compared to its neutral state, resulting in a smaller radius. Br- has gained an electron compared to its neutral state, increasing its size. F is in the same period as Br- but has a smaller atomic number, thus a smaller atomic radius. Mg has two more protons than F but is also in the same period, so its atomic radius is smaller.
Therefore, the order of increasing atomic radius is as follows: Mg < F < K+ < Br-.