This paper introduced the following mechanical problem: a rod was impacted by a particle with a certain velocity. After substituting the solutions, obtained by Direct Mode Shape Superposition Method (DMSM) available in the literatures, into the known condition that the initial non-dimensional velocity was equal to 1, the sum of the series, whose variable satisfied a certain constraining equation, was obtained. The conclusion was expressed in mathematical form. In the appendix, the programs were presented to test the conclusions obtained by using symbol computation software. The first step of the programs was to calculate the eigenvalues, and then substituted the values into the left side of the equations, which verified the correctness of the conclusions.