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In this paper we will extend the well-known chain of inequalities involving the Pythagorean means, namely the harmonic, geometric, and arithmetic means to the more refined chain of inequalities by including the logarithmic and identric means using nothing more than basic calculus. Of course, these results are all well-known and several proofs of them and their generalizations have been given. See [1-6] for more information. Our goal here is to present a unified approach and give the proofs as corollaries of one basic theorem.
roots of numbers have several interesting properties, most of which are
well-known. In this paper, our goal is to prove some lesser known results
concerning the digital roots of powers of numbers in an arithmetic progression.
We will also state some theorems concerning the digital roots of Fermat numbers
and star numbers. We will conclude our paper by an interesting application.