In this paper, we propose a physical protocol to verify the first nonzero term of a sequence using a deck of cards. The protocol lets a prover show the value of the first nonzero term of a given sequence to a verifier without revealing which term it is. Our protocol uses $Θ(1)$ shuffles, which is asymptotically lower than that of an existing protocol of Fukusawa and Manabe which uses $Θ(n)$ shuffles, where $n$ is the length of the sequence. We also apply our protocol to construct zero-knowledge proof protocols for three well-known logic puzzles: ABC End View, Goishi Hiroi, and Toichika. These protocols enables a prover to physically show that he/she know solutions of the puzzles without revealing them.