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Browse Classroom Capsules and Notes

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Displaying 1131 - 1140 of 1213

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.

A covering system is a system of \(k\) arithmetic progressions whose union includes all integers. This paper presents upper bounds on the number of consecutive integers which need to be checked...

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