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Why do these relationships exist?:
The higher the atomic number of an element means the atomic radii will be bigger because there are more electrons and shells/orbitals involved which will make the element bigger in size. The more electrons there are means the more shells there are which equals to a bigger atomic radii.
The first energisation energy will be lower the bigger the atomic number because a larger atomic number means more electrons. This means more shells and the electrons will be further away from the nucleus which equals to less nuclear attraction. This means as the electrons get further away from the nucleus (from the larger amount of electrons and more shells and shielding from the shells closer to the nucleus) there will be less attraction holding the electrons to the nucleus which will make it easier to remove.
I'm not sure what the next question means, but I hope this helps explain why the relationships exist :)
