Title: Spectral properties of Jacobi matrices and sum rules of special form. Abstract: We relate the properties of the elements of a Jacobi matrix from certain class to the properties of its spectral measure. The main tools we use are the so-called sum rules originally suggested by Case. Later, the sum rules were efficiently applied by Killip-Simon to the spectral analysis of Jacobi matrices. We use a modification of the method that permits us to work with sum rules of higher orders. As a corollary of the main theorem, we obtain a direct counterpart of a result of Molchanov-Novitskii-Vainberg for a ``continuous'' Schr\"odinger operator on a half-line.