In this thesis, it was aimed to introduce the telescopic numerical semigroups, which are a special symmetric numerical semigroup, in order to express the invariants of some telescopic numerical semigroups and the Betti numbers in terms of the generators of these semigroups and also to determine the catenary degrees of the obtained Betti numbers. In this study, firstly, the information about numerical semigroups, symmetric numerical semigroups, telescopic numerical semigroups, catenary degree and Betti numbers were given. Then some telescopic numerical semigroup families with embedding dimension three were given and some formulas were obtained for the Betti numbers of these families. By using these formulas, some results were found for the genus and Frobenius numbers of these families. Furthermore, some theorems were presented for the factorizations of the Betti numbers of these families. By using these theorems, some formulas were also derived for the catenary degrees of some of these Betti numbers.
