On the Number of Nonempty Subsets in a Given Set

Sets contain some numbers according to their properties and can help us write some series as a result of the numbers they contain. At this point, thanks to the equations arising from the mathematical series created at this point, the existence of some proofs in which generalizations can be expressed can be proved. In addition, these proofs can be tools or results in reaching new generalizations. In this direction, the aim of this study is to prove a sum sequence for the calculation of all subsets of a set other than the most empty set in which there are ordered sums of its elements except the empty set. In this direction, a different format of the summation series formed by subtracting the empty set from the number of all subsets that form the ordered sums of a set with n elements different from the empty set is obtained and the equality is proved by inductive proof method. As a result, the obtained general equality is presented as a new sum series. It is predicted that new generalizations can be reached thanks to this equality.