The author proves the following summation formula for a terminating \(_{3}F_{2}\) generalized hypergeometric series of unit argument \[ _{3}F_{2}\biggl(\begin{matrix} -n,\,a,\,c+1\\ b,\,c \\ \end{matrix},1 \biggr)= \frac{(b-a-1)_{n}\,(\alpha+1)_{n}}{(b)_{n}\,(\alpha)_{n}}, \] where \(\alpha=\frac{c(1+a-b)}{a-c}\). Two different proofs of this formula are given. Then using this formula the author obtains a reduction formula for the Kampé de Fériet (double generalized hypergeometric) function \(F_{q:2;0}^{p:2;0}[-x,x]\), which generalizes some earlier results dealing with transformation formulas for a \(_{2}F_{2}\) generalized hypergeometric function.