Summation on the basis of combinatorial representation of equal powers

In the paper the conclusion of combinatorial expressions for the sums of members of several sequences is considered. Conclusion is made on the basis of combinatorial representation of the sum of the weighted equal powers. The weighted members of a geometrical progression, the simple arithmetic-geometrical and combined progressions are subject to summation. One of principal places in the given conclusion occupies representation of members of each of the specified progressions in the form of matrix elements. The row of this matrix is formed with use of a gang of equal powers with the set weight factor. Besides, in work formulas of combinatorial identities with participation of free components of the sums of equal powers, and also separate power-member of sequence of equal powers or a geometrical progression are presented. All presented formulas have the general basis-components of the sums of equal powers.