Let p be an odd prime. The purpose of this paper is to refine methods of Ahlgren and Ono and Kilbourn in order to prove a general mod p^3 congruence for the Gaussian hypergeometric series {}_{n+1}F_{n}(\lambda) where n is an odd positive integer. As a result, we extend three recent supercongruences. The first is a result of Ono and Ahlgren on a supercongruence for Ap{\'e}ry numbers which was conjectured by Beukers in 1987. The second is one of Mortenson which relates truncated hypergeometric series to the number of F_{p} points of some family of Calabi-Yau manifolds. Finally, the third is a result of Loh and Rhodes on congruences between coefficients of modular forms corresponding to a particular class of elliptic curves and combinatorial objects. Additionally, we discuss the non-trivial methods of the computer summation package Sigma which were used to find explicit evaluations of two strange combinatorial identities involving generalized Harmonic sums.