Usually interpretated as a logical game that one can easily stumble upon on the internet, finding the general rule of a given sequence of terms is quite subjective, merely a gambling in finding the specific rule the author was thinking about when assigning the values. Therefore, even though one finds a “rule” that might apply for the given terms, if it results in a different answer than what was intended, then the solution is considered to be wrong, which is rather unfair. And there are infinitely such rules that could be found.
Thus, I will present the method that allows us to find such a “rule”, as well as the role it plays in more advanced problems of Algebra and Number Theory. This method is called ” The Lagrange Interpolation Formula”, which, paired with Chebyshev’s polynomials, offer unique solutions to less-known results that oftenly open pathways to easier alternatives for seemingly unapproachable olympiad problems.50_Lagrange_interpolation_and_chebyshev_polynomials