*By exploiting the geometry of the cobweb plot, the author provides a simple and elementary derivation of the parameter for the period-three cycle of the logistic map.*

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Displaying 1131 - 1140 of 1214

*The author gives a mapping of the ordered pairs of positive integers onto the positive integers which is immediately recognizable as a bijection.*

*The author uses Newton`s impact method to compute the centripetal force exerted on a particle moving uniformly on a circumference of a noneuclidean circle.*

*A combinatorial proof of a formula for the sum in the title is provided.*

*How do sequences of the form \((1+x/n)^{n+\alpha}\) with \(x >0\) approach their limits?*

*The author presents three solutions to a problem concerning the terms of a certain linear recurrence modulo prime numbers.*

*The number \(2\) is a quadratic residue mod \(p\) if \(p = 8k + 1\) or \(p = 8k + 7\), but not if \(p = 8k + 3\) or \(p = 8k + 5\). This is proved by a simple counting argument, assuming the...*

*Given a positive integer \(m\), the authors exhibit a group with the probability \(1/m\) that two randomly selected elements commute.*

*The number of black squares in a crossword puzzle, and the number of marks needed to bound the black squares allows one to determine the number of words the puzzle contains.*

*In this note an elementary proof of Stirling's asymptotic formula for \(n!\) is given.*