Why do melting points decrease down the group 1 and increade down the group 7? (core syllabus: Periodicity)

Elements in the group one (i.e. alkali metals from lithium to francium) exhibit metallic bonding: the positive nuclei are held together thanks to the attraction to delocalised electrons. As the number of electron shells increases down the group, and consequently the atomic radii get bigger, the attraction between the nuclei and outer shell (valence) electrons decreases. Hence, the bonds are weaker and less energy is required to break them.

Elements in the group seven (i.e. the halides form fluorine to iodine) normally form diatomic covalently bound molecules. The only type of interaction between them are the London dispersion forces (momentary dipole interactions). The magnitude of London forces is proportional to the mass of the molecules, so as the molecular weights of the halides increase down the group, more energy is required to break the interactions between them.