In this paper, we consider Sylvester's theorem on the largest prime divisor of a product
of consecutive terms of an arithmetic progression, and prove another generalization
of this theorem. As an application of this generalization, we provide an explicit
method to find perfect powers in a product of terms of binary recurrence sequences
and associated Lucas sequences whose indices come from consecutive terms of an arithmetic
progression. In particular, we prove explicit results for Fibonacci, Jacobsthal, Mersenne
and associated Lucas sequences.