The +1 oxidation state in group 13 and +2 oxidation state in group 14 becomes more and more stable with increasing atomic number. Explain.

The term inert pair effect is often used in relation to the increasing stability of oxidation states that are 2 less than the group valency for the heavier elements of groups 13, 14, 15 and 16. In group 13 all the elements show +3 oxidation state whereas Ga, In and Tl show +1 oxidation state also.
Boron, being small in size can lose its valence electrons to form B3+ ion and shows +3 oxidation state.
+1 oxidation state of Tl, Ga is due to inert pair effect. The outer shell s-electrons (ns2) penetrate to (n-1) d-electrons and thus becomes closer to nucleus and are more effectively puller towards the nucleus.
This results in less availability of ns2 electrons pair for bonding or ns2 electron pair becomes inert. This reluctance in the participation of ns2 electrons in bonding is termed as inert pair effect. The inert pair is more effective after n$\ge$4 and increases with increasing value of n.
For groups 14, in spite of four valence electrons, they do not form M4+ or M4- ionic compounds. They form covalent compounds with four bonds.
Ge, Sn and Pb also exhibit +2 oxidation state due to inert pair effect. Sn2+ and Pb2+ show ionic nature. The tendency to form +2 ionic state increases on moving down the group due to inert pair effect.